Yuk, kita mempelajari barisan geometri, deret geometri, dan deret geometri tak hingga! Seperti apa bentuknya dan bagaimana rumus-rumusnya? Simak artikel Matematika kelas 11 ini, ya!
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Jika kamu sudah membaca artikel tentang barisan dan deret aritmatika, kamu pastinya sudah tahu manfaat dari mempelajari konsep barisan dan deret dalam matematika. Nah, selain barisan dan deret aritmatika, ada satu lagi nih, yang mau kita bahas di artikel ini, yaitu barisan dan deret geometri.
Apa itu barisan dan deret geometri? Apa sih, perbedaannya dengan barisan dan deret aritmatika? Oke, supaya kamu nggak bingung, yuk langsung baca penjelasannya di bawah ini!
Barisan geometri adalah pola yang memiliki pengali atau rasio yang tetap untuk setiap 2 suku yang berdekatan. Rasio pada barisan geometri biasa disimbolkan dengan r. Barisan geometri juga biasa disebut sebagai barisan ukur.
Contoh lebih mudahnya begini, misal kamu punya barisan seperti ini:
1, 3, 9, 27, …
Dari barisan tersebut, kita bisa lihat antara suku pertama dengan suku kedua, antara suku kedua dan suku ketiga dan seterusnya selalu punya pengali yang tetap, yaitu 3. Dengan demikian, barisan ini termasuk barisan geometri.
Nah, kalau barisan ini dituliskan dalam bentuk penjumlahan, namanya jadi deret geometri. Deret geometri itu bentuk penjumlahan dari barisan geometri. Penulisannya adalah seperti ini:
1 + 3 + 9 + 27 + …
Paham ya, bedanya barisan dan deret? Lalu, kalau deret geometri tak hingga itu apa?
Deret geometri tak hingga hampir sama dengan deret geometri, namun deret tersebut diteruskan hingga nilainya tak hingga. Nanti kita bahas lebih lanjut ya, supaya kamu bisa lebih paham. Sekarang, kita bahas mulai dari barisan dan deret geometri dulu, yuk! Lalu selanjutnya kita akan bahas tentang deret geometri tak hingga.
Tadi, kita sudah mengenal pengertian serta contoh dari barisan geometri dan deret geometri. Sekarang, kita belajar rumus-rumusnya, ya!
Pada barisan geometri dan deret geometri, terdapat tiga rumus yang harus kamu ketahui, yaitu rumus rasio, rumus Un, dan rumus Sn. Kita bahas satu per satu, ya!
1. Rumus Rasio (r) pada Barisan dan Deret Geometri
Rasio adalah nilai pengali pada barisan dan deret. Rumus untuk mencari rasio pada barisan geometri dan deret geometri adalah seperti infografis berikut.
Misalnya kita punya barisan geometri:
1, 3, 9, 27, 81, ….
Suku pertama (a) dari barisan geometri tersebut adalah 1. Maka r-nya adalah:
Jadi, rasio dari barisan geometri tersebut adalah 3.
Sekarang kita pelajari rumus suku ke–n (Un), yuk!
2. Rumus Suku ke-n (Un) pada Barisan dan Deret Geometri
Un adalah suku ke-n pada barisan dan deret. Untuk mencari Un pada barisan geometri dan deret geometri, kamu bisa menggunakan rumus berikut ini.
Misalnya kita punya barisan geometri:
1, 3, 9, 27, 81, ….
Lalu, kita coba cari Un nya. Misalnya n yang mau dicari adalah 6, maka:
Un = arn-1
U6 = ar5
U6 = 1 . 35
U6 = 1 . 243
U6 = 243
Jadi, U6 dari barisan geometri tersebut adalah 243.
Mudah kan, rumusnya? Syaratnya adalah kamu harus mengetahui berapa nilai a dan r-nya. Dengan begitu, kamu sudah bisa mencari Un dengan mudah. Sekarang, kita cari tahu rumus selanjutnya yuk!
3. Rumus Jumlah Suku ke-n (Sn) pada Barisan dan Deret Geometri
Sn adalah jumlah suku ke-n pada barisan dan deret. Nah, bagaimana cara kita mencari tau Sn pada barisan geometri dan deret geometri? Berikut ini adalah rumusnya. Check it out!
Misalnya kita punya barisan geometri:
1, 3, 9, 27, 81, ….
Lalu, kita coba cari Sn nya. Misalnya n yang mau dicari adalah 3, maka:
Jadi, S3 dari barisan geometri tersebut adalah 13.
Oke, itu dia rumus Sn dalam barisan geometri dan deret geometri. Nah sekarang, kita lanjut bahas tentang deret geometri tak hingga, yuk!
Baca Juga: Barisan Aritmatika Bertingkat: Konsep Dasar, Rumus & Contoh Soal
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Gimana? Apakah kamu sudah mulai paham dengan konsep barisan dan deret geometri di atas? Kalo masih ada poin-poin yang belum kamu mengerti, mending belajar sama ahlinya, deh. Belajar bareng kakak-kakak pengajar di Ruangguru Privat Matematika misalnya.
Belajar nggak cuma menyenangkan, tapi kamu juga bakal diajari konsepnya sampai paham! Para pengajar di Ruangguru Privat juga sudah terstandarisasi kualitasnya, loh. Kamu juga bisa pilih nih, mau diajarkan secara langsung (offline) atau daring (online). Fleksibel, kan? Untuk info lebih lanjut, cuss klik link berikut!
Konsep Deret Geometri Tak Hingga
Deret geometri tak hingga itu dibagi menjadi 2 jenis yaitu deret geometri tak hingga divergen dan deret geometri tak hingga konvergen. Keduanya memiliki perbedaan yang cukup penting.
Yuk, kita lihat pengertian dari kedua jenis deret geometri tak hingga tersebut beserta perbedaannya!
1. Deret Geometri Tak Hingga Divergen
Deret geometri tak hingga divergen adalah suatu deret yang nilai bilangannya semakin membesar dan tidak bisa dihitung jumlahnya. Bisa kita lihat seperti di bawah ini,
1 + 3 + 9 + 27 + 81 + ……………
Kalau ditanya berapa sih, jumlah seluruhnya? Jumlah seluruhnya tidak bisa dihitung karena nilainya semakin besar.
2. Deret Geometri Tak Hingga Konvergen
Berbeda dengan deret geometri tak hingga divergen, deret geometri tak hingga konvergen merupakan suatu deret di mana nilai bilangannya semakin mengecil dan dapat dihitung jumlahnya. Seperti di bawah ini:
Semakin lama nilainya semakin mengecil dan ujungnya akan mendekati angka 0. Hal ini membuat deret geometri tak hingga konvergen dapat dihitung jika ditanyakan jumlah seluruhnya.
Lalu bagaimana cara menghitung jumlah seluruhnya dari deret geometri tak hingga konvergen?
3. Rumus Stak hingga pada Deret Geometri Tak Hingga Konvergen
Sebelum masuk ke rumus, ada syarat terlebih dahulu jika kamu bertemu dengan deret geometri tak hingga konvergen, yaitu rasionya harus bernilai antara -1 sampai 1 (-1 > r > 1) dan ini berlaku untuk negatif dan positif. Contohnya seperti deret di atas. Deret di atas rasionya adalah sehingga bisa dihitung jumlah tak hingganya.
Nah, sekarang kita lihat yuk rumus untuk menghitung Stak hingga atau jumlah tak hingganya!
Misalnya kita punya deret geometri tak hingga konvergen:
Lalu, kita coba cari Stak hingga nya, maka:
Jadi, Stak hingga darideret geometri tak hingga konvergen tersebut adalah .
Itu dia penjelasan tentang barisan geometri, deret geometri, serta deret geometri tak hingga. Bagaimana, teman-teman? Kamu sudah paham, kan? Atau kamu masih belum puas dengan penjelasannya? Hmm tenang, kamu bisa nih, belajar melalui video animasi di ruangbelajar.
Di sana, kamu bisa belajar sekaligus latihan soal-soal. Selain itu, waktu belajar kamu akan lebih efektif, dan tidak akan menyita waktu bermain kamu. Jadiii tunggu apa lagi? Buruan download aplikasi ruangguru!
Referensi:
Wirodikromo, S. dan Darmanto, M. (2019) Matematika untuk SMA/MA Kelas XI kelompok Wajib 2. Jakarta: Erlangga.
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<h1 class="content-title">
Konsep Barisan & Deret Geometri, Rumus, serta Contoh Soal | Matematika Kelas 11 </h1>
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<p class="content-author"><a href="https://www.ruangguru.com/blog/author/kenya-swawikanti" class="">Kenya Swawikanti</a> </p>
<p class="content-info"><span>November 25, 2024 • </span> <span>6 minutes read</span></p>
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<p><img decoding="async" style="width: 820px; margin-left: auto; margin-right: auto; display: block;" src="https://cdn-web.ruangguru.com/landing-pages/assets/hs/Rumus-Rumus%20Barisan%20dan%20Deret%20Geometri.jpg" alt="Rumus-Rumus Barisan dan Deret Geometri" width="820" /></p>
<blockquote>
<p style="font-size: 16px; text-align: center;"><em>Yuk, kita mempelajari <strong>barisan geometri, deret geometri, dan deret geometri tak hingga</strong>! Seperti apa bentuknya dan bagaimana rumus-rumusnya? Simak <strong><a href="https://www.ruangguru.com/blog/c/matematika/matematika-sma-kelas-11">artikel Matematika kelas 11</a></strong> ini, ya!</em></p>
<p style="text-align: center; font-size: 16px;">—</p>
</blockquote>
<p> </p>
<p style="text-align: justify;">Jika kamu sudah membaca artikel tentang <strong><a href="/blog/manfaat-belajar-barisan-dan-deret-aritmatika" rel="noopener">barisan dan deret aritmatika</a></strong>, kamu pastinya sudah tahu manfaat dari mempelajari konsep barisan dan deret dalam matematika. Nah, selain barisan dan deret aritmatika, ada satu lagi nih, yang mau kita bahas di artikel ini, yaitu barisan dan deret geometri.</p>
<p style="text-align: justify;"><span style="font-weight: normal;">Apa itu barisan dan deret geometri?</span> Apa sih, perbedaannya dengan barisan dan deret aritmatika? Oke, supaya kamu nggak bingung, yuk langsung baca penjelasannya di bawah ini!</p>
<p style="text-align: justify;"><span style="font-weight: bold;">Barisan geometri adalah pola yang memiliki pengali atau rasio yang tetap untuk setiap 2 suku yang berdekatan</span>. Rasio pada barisan geometri biasa disimbolkan dengan r. Barisan geometri juga biasa disebut sebagai barisan ukur.</p>
<p style="text-align: justify;">Contoh lebih mudahnya begini, misal kamu punya barisan seperti ini:</p>
<p style="font-weight: bold; text-align: center;"><span style="font-size: 14pt;">1, 3, 9, 27, …</span></p>
<p> </p>
<p style="text-align: justify;">Dari barisan tersebut, kita bisa lihat antara suku pertama dengan suku kedua, antara suku kedua dan suku ketiga dan seterusnya selalu punya <span style="font-weight: bold;">pengali yang tetap, yaitu 3</span>. Dengan demikian, barisan ini termasuk <span style="font-weight: bold;">barisan geometri</span>.</p>
<p style="text-align: justify;">Nah, kalau barisan ini dituliskan dalam bentuk penjumlahan, namanya jadi <span style="font-weight: normal;">deret geometri</span>. <span style="font-weight: bold;">Deret geometri itu bentuk penjumlahan dari barisan geometri</span>. Penulisannya adalah seperti ini:</p>
<p style="text-align: center; font-weight: bold;"><span style="font-size: 14pt;">1 + 3 + 9 + 27 + …</span></p>
<p> </p>
<p style="text-align: justify;">Paham ya, bedanya barisan dan deret? Lalu, kalau deret geometri tak hingga itu apa?</p>
<p style="text-align: justify;">Deret geometri tak hingga hampir sama dengan deret geometri, namun deret tersebut diteruskan hingga nilainya tak hingga. Nanti kita bahas lebih lanjut ya, supaya kamu bisa lebih paham. Sekarang, kita bahas mulai dari barisan dan deret geometri dulu, yuk! Lalu selanjutnya kita akan bahas tentang deret geometri tak hingga.</p>
<p> </p>
<h2><span style="font-size: 18pt;">Konsep Barisan Geometri dan Deret Geometri</span></h2>
<p style="text-align: justify;">Tadi, kita sudah mengenal pengertian serta contoh dari barisan geometri dan deret geometri. Sekarang, kita belajar rumus-rumusnya, ya!</p>
<p style="text-align: justify;">Pada barisan geometri dan deret geometri, terdapat tiga rumus yang harus kamu ketahui, yaitu <span style="font-weight: bold;">rumus rasio, rumus U<sub>n</sub>, dan rumus S<sub>n</sub></span>. Kita bahas satu per satu, ya!</p>
<p> </p>
<h3><span style="font-size: 14pt;">1. Rumus Rasio (r) pada Barisan dan Deret Geometri</span></h3>
<p style="text-align: justify;">Rasio adalah <span style="font-weight: bold;">nilai pengali pada barisan dan deret. </span>Rumus untuk mencari <span style="font-weight: normal;">rasio </span>pada barisan geometri dan deret geometri adalah seperti infografis berikut.</p>
<p style="text-align: justify;"><img decoding="async" style="width: 600px; margin-left: auto; margin-right: auto; display: block;" src="https://cdn-web.ruangguru.com/landing-pages/assets/hs/Rumus%20Rasio%20Baris%20dan%20Deret%20Geometri.jpg" alt="Rumus Rasio Baris dan Deret Geometri" width="600" /></p>
<p> </p>
<p style="text-align: justify;">Misalnya kita punya barisan geometri:</p>
<p style="text-align: justify;">1, 3, 9, 27, 81, ….</p>
<p style="text-align: justify;"><span style="font-weight: bold;">Suku pertama</span> <span style="font-weight: bold;">(a)</span> dari barisan geometri tersebut adalah <span style="font-weight: bold;">1</span>. Maka r-nya adalah:</p>
<p style="text-align: justify;"><img decoding="async" style="width: 136px;" src="https://cdn-web.ruangguru.com/landing-pages/assets/hs/mtk%201.png" alt="mtk 1" width="136" /></p>
<p style="text-align: justify;"><span style="background-color: transparent; font-size: 1rem;">Jadi, <span style="font-weight: bold;">rasio</span> dari barisan geometri tersebut adalah <span style="font-weight: bold;">3</span>. </span></p>
<p style="text-align: justify;"><span style="background-color: transparent; font-size: 1rem;">Sekarang kita pelajari rumu</span><span style="background-color: transparent; font-size: 1rem;">s</span> <span style="background-color: transparent; font-size: 1rem;">s</span><span style="background-color: transparent; font-size: 1rem;">uku ke–n (U<sub>n</sub></span><span style="background-color: transparent; font-size: 1rem;">), yuk!</span></p>
<p> </p>
<h3><span style="font-size: 14pt;">2. Rumus Suku ke-n (U<sub>n</sub>) pada Barisan dan Deret Geometri</span></h3>
<p style="text-align: justify;">U<sub>n </sub>adalah <span style="font-weight: bold;">suku ke-n pada barisan dan deret</span>. Untuk mencari U<sub>n</sub> pada barisan geometri dan deret geometri, kamu bisa menggunakan rumus berikut ini.</p>
<p style="text-align: justify;"><img decoding="async" style="width: 600px; margin-left: auto; margin-right: auto; display: block;" src="https://cdn-web.ruangguru.com/landing-pages/assets/hs/Rumus%20Un%20Baris%20dan%20Deret%20Geometri.jpg" alt="Rumus Un Barisan dan Deret Geometri" width="600" /></p>
<p> </p>
<p style="text-align: justify;">Misalnya kita punya barisan geometri:</p>
<p style="text-align: justify;">1, 3, 9, 27, 81, ….</p>
<p style="text-align: justify;">Lalu, kita coba cari <span style="font-weight: bold;">U</span><sub><span style="font-weight: bold;">n</span> </sub>nya. Misalnya <span style="font-weight: bold;">n</span> yang mau dicari adalah <span style="font-weight: bold;">6</span>, maka:</p>
<p style="text-align: justify;">U<sub>n</sub> = ar<sup>n-1</sup></p>
<p style="text-align: justify;">U<sub>6</sub> = ar<sup>5</sup></p>
<p style="text-align: justify;">U<sub>6</sub> = 1 . 3<sup>5</sup></p>
<p style="text-align: justify;">U<sub>6 </sub>= 1 . 243</p>
<p style="text-align: justify;">U<sub>6 </sub>= 243</p>
<p style="text-align: justify;">Jadi, <span style="font-weight: bold;">U<sub>6</sub></span> dari barisan geometri tersebut adalah <span style="font-weight: bold;">243</span>.</p>
<p style="text-align: justify;">Mudah kan, rumusnya? Syaratnya adalah kamu harus mengetahui berapa nilai a dan r-<span style="background-color: transparent;">nya</span><span style="background-color: transparent;">. Dengan begitu, kamu sudah bisa mencari U<sub>n</sub> dengan mudah. Sekarang, kita cari tahu rumus selanjutnya yuk!</span></p>
<p> </p>
<h3><span style="font-size: 14pt;">3. Rumus Jumlah Suku ke-n (S<sub>n</sub>) pada Barisan dan Deret Geometri</span></h3>
<p style="text-align: justify;">S<sub>n</sub> adalah<span style="font-weight: bold;"> jumlah suku ke-n pada barisan dan deret</span>. Nah, bagaimana cara kita mencari tau S<sub>n</sub> pada barisan geometri dan deret geometri? Berikut ini adalah rumusnya. <em>Check it out!</em></p>
<p style="text-align: justify;"><img decoding="async" style="width: 600px; margin-left: auto; margin-right: auto; display: block;" src="https://cdn-web.ruangguru.com/landing-pages/assets/hs/Rumus%20Sn%20Baris%20dan%20Deret%20Geometri.jpg" alt="Rumus Sn Barisan dan Deret Geometri" width="600" /></p>
<p> </p>
<p style="text-align: justify;">Misalnya kita punya barisan geometri:</p>
<p style="text-align: justify;">1, 3, 9, 27, 81, ….</p>
<p style="text-align: justify;">Lalu, kita coba cari <span style="font-weight: bold;">S</span><sub><span style="font-weight: bold;">n</span> </sub>nya. Misalnya <span style="font-weight: bold;">n</span> yang mau dicari adalah <span style="font-weight: bold;">3</span>, maka:</p>
<p style="text-align: justify;"><img decoding="async" style="width: 200px;" src="https://cdn-web.ruangguru.com/landing-pages/assets/hs/mtk%202.png" alt="mtk 2" width="200" /></p>
<p style="text-align: justify;">Jadi, <span style="font-weight: bold;">S</span><sub><span style="font-weight: bold;">3</span> </sub>dari barisan geometri tersebut adalah <span style="font-weight: bold;">13</span>.</p>
<p style="text-align: justify;">Oke, itu dia rumus S<sub>n</sub> dalam barisan geometri dan deret geometri. Nah sekarang, kita lanjut bahas tentang deret geometri tak hingga, yuk!</p>
<p style="text-align: justify; font-weight: bold;">Baca Juga: <a href="https://www.ruangguru.com/blog/barisan-aritmatika-bertingkat-konsep-dasar-rumus-dan-contoh-soal">Barisan Aritmatika Bertingkat: Konsep Dasar, Rumus & Contoh Soal</a></p>
<p style="text-align: center;">—</p>
<p style="text-align: justify;">Gimana? Apakah kamu sudah mulai paham dengan konsep barisan dan deret geometri di atas? Kalo masih ada poin-poin yang belum kamu mengerti, mending belajar sama ahlinya, deh. Belajar bareng kakak-kakak pengajar di <a href="https://www.ruangguru.com/privat/matematika"><strong>Ruangguru Privat Matematika</strong></a> misalnya.</p>
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<h2><span style="font-size: 18pt;">Konsep Deret Geometri Tak Hingga</span></h2>
<p style="text-align: justify;">Deret geometri tak hingga itu dibagi menjadi 2 jenis yaitu <span style="font-weight: bold;">deret geometri tak hingga divergen</span> dan <span style="font-weight: bold;">deret geometri tak hingga konvergen</span>. Keduanya memiliki perbedaan yang cukup penting.</p>
<p style="text-align: justify;">Yuk, kita lihat pengertian dari kedua jenis deret geometri tak hingga tersebut beserta perbedaannya!</p>
<p> </p>
<h3><span style="font-size: 14pt;">1. Deret Geometri Tak Hingga Divergen</span></h3>
<p style="text-align: justify;">Deret geometri tak hingga divergen adalah suatu deret yang <span style="font-weight: bold;">nilai bilangannya</span> <span style="font-weight: bold;">semakin membesar</span> <span style="font-weight: bold;">dan tidak bisa dihitung jumlahnya</span>. Bisa kita lihat seperti di bawah ini,</p>
<p style="text-align: justify;"><span style="font-size: 14pt;">1 + 3 + 9 + 27 + 81 + ……………</span></p>
<p> </p>
<p style="text-align: justify;">Kalau ditanya berapa sih, jumlah seluruhnya? Jumlah seluruhnya tidak bisa dihitung karena nilainya semakin besar.</p>
<p> </p>
<h3><span style="font-size: 14pt;">2. Deret Geometri Tak Hingga Konvergen</span></h3>
<p style="text-align: justify;">Berbeda dengan deret geometri tak hingga divergen, deret geometri tak hingga konvergen merupakan suatu deret di mana <span style="font-weight: bold;">nilai bilangannya semakin mengecil dan dapat dihitung jumlahnya</span>. Seperti di bawah ini:</p>
<p style="text-align: justify;"><img decoding="async" style="width: 527px;" src="https://cdn-web.ruangguru.com/landing-pages/assets/hs/mtk%203.png" alt="mtk 3" width="527" /></p>
<p style="text-align: justify;"><span style="background-color: transparent;">Semakin lama nilainya semakin mengecil dan ujungnya akan mendekati angka 0. Hal ini membuat deret geometri tak hingga konvergen <span style="font-weight: bold;">dapat dihitung</span> jika ditanyakan jumlah seluruhnya.</span></p>
<p style="text-align: justify;">Lalu bagaimana cara menghitung jumlah seluruhnya dari deret geometri tak hingga konvergen?</p>
<p> </p>
<h3><span style="font-size: 14pt;">3. Rumus Stak hingga pada Deret Geometri Tak Hingga Konvergen</span></h3>
<p style="text-align: justify;">Sebelum masuk ke rumus, ada syarat terlebih dahulu jika kamu bertemu dengan deret geometri tak hingga konvergen, yaitu <span style="font-weight: bold;">rasionya harus bernilai antara -1 sampai 1</span> <span style="font-weight: bold;">(-1 > r > 1)</span> dan ini <span style="font-weight: bold;">berlaku untuk negatif dan positif</span>. Contohnya seperti deret di atas. Deret di atas rasionya adalah <img decoding="async" style="width: 62px;" src="https://cdn-web.ruangguru.com/landing-pages/assets/hs/mtk%204.png" alt="mtk 4" width="62" /> sehingga bisa dihitung jumlah tak hingganya.</p>
<p style="text-align: justify;">Nah, sekarang kita lihat yuk rumus untuk menghitung S<sub>tak hingga </sub>atau jumlah tak hingganya!</p>
<p style="text-align: justify;"><img decoding="async" style="width: 600px; margin-left: auto; margin-right: auto; display: block;" src="https://cdn-web.ruangguru.com/landing-pages/assets/hs/Rumus%20S%20Tak%20Hingga%20Baris%20dan%20Deret%20Geometri.jpg" alt="Rumus S Tak Hingga Barisan dan Deret Geometri" width="600" /></p>
<p> </p>
<p style="text-align: justify;">Misalnya kita punya deret geometri tak hingga konvergen:</p>
<p style="text-align: justify;"><img decoding="async" style="width: 527px;" src="https://cdn-web.ruangguru.com/landing-pages/assets/hs/mtk%203.png" alt="mtk 3" width="527" /></p>
<p style="text-align: justify;">Lalu, kita coba cari <span style="font-weight: bold;">S<sub>tak hingga</sub></span><sub> </sub>nya, maka:</p>
<p style="text-align: justify;"><img decoding="async" style="width: 206px;" src="https://cdn-web.ruangguru.com/landing-pages/assets/hs/mtk%205.png" alt="mtk 5" width="206" /></p>
<p style="text-align: justify;">Jadi, <span style="font-weight: bold;">S<sub>tak hingga</sub></span><sub> </sub>darideret geometri tak hingga konvergen tersebut adalah <span style="font-weight: bold;"><img decoding="async" style="width: 39px;" src="https://cdn-web.ruangguru.com/landing-pages/assets/hs/mtk%206.png" alt="mtk 6" width="39" /></span>.</p>
<p style="text-align: justify;"><span style="background-color: transparent;">Itu dia penjelasan tentang barisan geometri, deret geometri, serta deret geometri tak hingga. Bagaimana, teman-teman? Kamu sudah paham, kan? Atau kamu masih belum puas dengan penjelasannya? </span><span style="background-color: transparent;"><em>Hmm</em> tenang, kamu bisa nih, belajar melalui video animasi di </span><a style="font-weight: bold;" href="https://www.ruangguru.com/ruangbelajar" target="_blank" rel="noopener">ruangbelajar</a><span style="background-color: transparent;">. </span></p>
<p style="text-align: justify;"><span style="background-color: transparent;">Di sana, kamu bisa belajar sekaligus latihan soal-soal. Selain itu, waktu belajar kamu akan lebih efektif, dan tidak akan menyita waktu bermain kamu. Jadiii tunggu apa lagi? Buruan <em>download </em><span style="font-weight: normal;">aplikasi ruangguru!</span></span></p>
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<p><strong>Referensi:</strong></p>
<div style="text-align: justify;">
<p style="text-align: left;">Wirodikromo, S. dan Darmanto, M. (2019) <em>Matematika untuk SMA/MA Kelas XI kelompok Wajib 2</em>. Jakarta: Erlangga.</p>
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<div class="d-lg-none text-center">
<nav class="footer-menu social">
<p class="title">Ikuti Kami</p>
<a target="_blank" href="https://www.instagram.com/ruangguru/">
<picture>
<source srcset="https://cdn-web-2.ruangguru.com/static/instagram.png?height=60&convert=webp" type="image/webp">
<source srcset="https://cdn-web-2.ruangguru.com/static/instagram.png?height=60" type="image/png">
<img src="https://cdn-web-2.ruangguru.com/static/instagram.png" loading="lazy" alt="Logo Instagram Footer" width="32px" height="32px" />
</picture>
</a>
<a target="_blank" href="https://www.facebook.com/ruanggurucom/">
<picture>
<source srcset="https://cdn-web-2.ruangguru.com/static/facebook.png?height=60&convert=webp" type="image/webp">
<source srcset="https://cdn-web-2.ruangguru.com/static/facebook.png?height=60" type="image/png">
<img src="https://cdn-web-2.ruangguru.com/static/facebook.png" loading="lazy" alt="Logo Facebook Footer" width="32px" height="32px" />
</picture>
</a>
<a target="_blank" href="https://twitter.com/ruangguru/">
<picture>
<source srcset="https://cdn-web-2.ruangguru.com/static/twitter.png?height=60&convert=webp" type="image/webp">
<source srcset="https://cdn-web-2.ruangguru.com/static/twitter.png?height=60" type="image/png">
<img src="https://cdn-web-2.ruangguru.com/static/twitter.png" loading=" lazy" alt="Logo Twitter Footer" width="32px" height="32px" />
</picture>
</a>
<a target="_blank" href="https://www.youtube.com/ruangguru">
<picture>
<source srcset="https://cdn-web-2.ruangguru.com/static/youtube.png?height=60&convert=webp" type="image/webp">
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<img src="https://cdn-web-2.ruangguru.com/static/youtube.png" loading=" lazy" alt="Logo Youtube Footer" width="32px" height="32px" />
</picture>
</a>
<a target="_blank" href="https://www.linkedin.com/company/ruangguru-com">
<picture>
<source srcset="https://cdn-web-2.ruangguru.com/static/linkedin.png?height=60&convert=webp" type="image/webp">
<source srcset="https://cdn-web-2.ruangguru.com/static/linkedin.png?height=60" type="image/png">
<img src="https://cdn-web-2.ruangguru.com/static/linkedin.png" loading=" lazy" alt="Logo LinkedIn Footer" width="32px" height="32px" />
</picture>
</a>
<a target="_blank" href="https://www.tiktok.com/@ruangguru">
<picture>
<source srcset="https://cdn-web-2.ruangguru.com/static/tiktok.png?height=60&convert=webp" type="image/webp">
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<img src="https://cdn-web-2.ruangguru.com/static/tiktok.png" loading=" lazy" alt="Logo TikTok Footer" width="32px" height="32px" />
</picture>
</a>
</nav>
</div>
<p class="footer-copyright">
© 2026 All Rights Reserved PT. Ruang Raya Indonesia
</p>
</div>
<style>
.bantuan-container {
position: relative;
}
.bantuan-element {
position: fixed;
z-index: 999;
display: flex;
align-items: flex-end;
justify-content: flex-end;
cursor: pointer;
transition: all 0.6s ease;
}
.bantuan-top {
bottom: 11rem;
right: 2rem;
}
.bantuan-bottom {
bottom: 2rem;
right: 2rem;
}
.bantuan-top-2 {
bottom: 15rem;
right: 2rem;
}
.bantuan-top-3 {
bottom: 19rem;
right: 2rem;
}
.bantuan-bottom-2 {
bottom: 5rem;
right: 2rem;
}
.bantuan-bottom-3 {
bottom: 8rem;
right: 2rem;
}
@media (min-width: 768px) {
.bantuan-element img {
width: auto;
transition: all 0.4s ease;
}
.bantuan-top img {
width: 140px;
}
.bantuan-bottom img {
width: 140px;
}
.bantuan-top-2 img,
.bantuan-top-3 img {
width: 140px;
}
.bantuan-bottom-2 img,
.bantuan-bottom-3 img {
width: 140px;
}
.bantuan-element.clicked {
bottom: 9rem;
right: 2.5rem;
}
.bantuan-element.clicked img {
width: auto;
height: 64px;
}
}
@media (max-width: 767px) {
.bantuan-top {
bottom: 8rem;
right: 1rem;
align-items: center;
}
.bantuan-bottom {
bottom: 2rem;
right: 1.25rem;
align-items: center;
}
.bantuan-top-2 {
bottom: 12rem;
right: 1rem;
align-items: center;
}
.bantuan-top-3 {
bottom: 16rem;
right: 1rem;
align-items: center;
}
.bantuan-bottom-2 {
bottom: 5rem;
right: 1.25rem;
align-items: center;
}
.bantuan-bottom-3 {
bottom: 8rem;
right: 1.25rem;
align-items: center;
}
.bantuan-element img {
width: auto;
}
.bantuan-top img,
.bantuan-top-2 img,
.bantuan-top-3 img {
width: 96px;
}
.bantuan-bottom img,
.bantuan-bottom-2 img,
.bantuan-bottom-3 img {
width: 96px;
}
}
.bantuan-element:hover {
transform: scale(1.05);
}
.bantuan-element img {
max-width: 100%;
height: auto;
}
</style>
<div id="bantuan-divs-container" style="position: relative" data-product="ruangguru" data-table="rg_floating_icon"></div>
<script>
class FloatingDivsComponent {
constructor(product = 'ruangguru', table = 'rg_floating_icon') {
this.container = document.getElementById('bantuan-divs-container');
this.floatingElements = [];
this.product = product;
this.table = table;
// Configuration
this.config = {
baseApiUrl: 'https://gw.ruangguru.com/api/v3/rdb/p_ddgy8hjx5dxsqx/rg_banner/views/',
apiKey: 'vLyK3dJrDBRmOwueX2OJXibMkgsUbBk0WS4ua2p1',
maxElements: 2,
zIndexBase: 100,
};
// Build dynamic API URL based on table parameter
this.config.apiUrl = this.buildApiUrl();
this.init();
}
buildApiUrl() {
const baseUrl = this.config.baseApiUrl;
const tableName = this.table || 'rg_floating_icon';
const queryParams =
'offset=0&limit=99999&sort=rankPosition&where=(type,eq,floating)~and(showIn,anyof,blog)~and(hide,is,false)~and(product,eq,' +
this.product +
')&nested%5Bcta%5D%5Bfields%5D=Id,name,ctaId,imageUrl,ctaLink,targetUrl';
return `${baseUrl}${tableName}?${queryParams}`;
}
async init() {
try {
await this.fetchFloatingElements();
this.renderFloatingElements();
} catch (error) {
console.error('Failed to initialize floating divs component:', error);
}
}
async fetchFloatingElements() {
try {
const options = {
method: 'GET',
headers: {
'xc-token': this.config.apiKey || 'vLyK3dJrDBRmOwueX2OJXibMkgsUbBk0WS4ua2p1',
},
};
const urlNoco = this.config.apiUrl;
const response = await fetch(urlNoco, options);
if (!response.ok) {
throw new Error(`HTTP error! status: ${response.status}`);
}
const data = await response.json();
this.floatingElements = this.processApiResponse(data);
} catch (error) {
console.error('Error fetching floating elements:', error);
throw error;
}
}
processApiResponse(data) {
if (!data || !data.list || !Array.isArray(data.list)) {
return [];
}
console.log(data.list);
// Ensure we only get maximum 2 elements
const limitedElements = data.list.slice(0, this.config.maxElements);
return limitedElements;
}
renderFloatingElements() {
if (!this.container || this.floatingElements.length === 0) {
return;
}
// Additional safeguard: ensure we never render more than maxElements
const elementsToRender = this.floatingElements.slice(0, this.config.maxElements);
// Clean up existing elements and their event listeners
this.cleanupExistingElements();
// Clear existing content
this.container.innerHTML = '';
elementsToRender.forEach((element, index) => {
const floatingDiv = this.createFloatingDiv(element, index);
this.container.appendChild(floatingDiv);
});
}
cleanupExistingElements() {
// Clear existing elements (no need for event listener cleanup with CSS approach)
const existingElements = this.container.querySelectorAll('[data-floating-element]');
existingElements.forEach(element => {
element.remove();
});
}
createFloatingDiv(element, index) {
const div = document.createElement('div');
const cta = element.cta[0]; // Use the first CTA
// Add data attribute for cleanup tracking
div.setAttribute('data-floating-element', 'true');
// Apply CSS classes based on rankPosition
const positionClass = this.getPositionClass(element.rankPosition, index);
div.className = `bantuan-element ${positionClass}`;
div.style.zIndex = this.config.zIndexBase + index;
// Create the link and image
const link = document.createElement('a');
link.href = 'https://cta.ruangguru.com/' + cta.ctaId || '#';
link.target = '_blank';
// Add tracking attributes
link.setAttribute('data-tracking-event-type', 'landing_page__core__item_click');
link.setAttribute(
'data-tracking-context',
JSON.stringify({
product_name: this.product,
cta_action: 'redirection',
cta_action_detail: link.href,
item_type: 'floating',
item_name: element.name || 'Floating element',
item_category: 'payment',
item_position: index,
item_serial: null,
section_type: 'floating_item',
section_position: 'body',
section_name: 'floating_item',
})
);
const img = document.createElement('img');
img.src = element.imageUrl + '?h=200&convert=webp';
img.alt = element.name || 'Floating element';
img.loading = 'lazy';
// Add click tracking
link.addEventListener('click', () => {
this.trackClick(element, cta);
});
link.appendChild(img);
div.appendChild(link);
return div;
}
getPositionClass(rankPosition, index) {
// Define positioning classes based on rankPosition
if (rankPosition === 1) {
return 'bantuan-top'; // Top position
} else if (rankPosition === 2) {
return 'bantuan-bottom'; // Bottom position
} else {
// For additional elements, distribute them evenly
return `bantuan-${index % 2 === 0 ? 'top' : 'bottom'}-${Math.floor(index / 2) + 1}`;
}
}
trackClick(element, cta) {
// Implement click tracking logic here
console.log('Floating element clicked:', {
elementId: element.Id,
elementName: cta.name,
ctaId: cta.ctaId,
targetUrl: cta.targetUrl,
});
}
// Public method to manually refresh
async refresh() {
try {
await this.fetchFloatingElements();
this.renderFloatingElements();
} catch (error) {
console.error('Failed to refresh floating elements:', error);
throw error;
}
}
// Public method to get current floating elements
getFloatingElements() {
return [...this.floatingElements];
}
}
// Initialize the component when DOM is ready
document.addEventListener('DOMContentLoaded', () => {
// Get product and table parameters from data attributes
const container = document.getElementById('bantuan-divs-container');
const product = container.getAttribute('data-product') || 'ruangguru';
const table = container.getAttribute('data-table') || 'rg_floating_icon';
new FloatingDivsComponent(product, table);
});
// Export for use in other modules if needed
if (typeof module !== 'undefined' && module.exports) {
module.exports = FloatingDivsComponent;
}
</script> <div id="wp-install-nudge" style="
position: fixed;
bottom: 0rem;
right: 0;
z-index: 999;
cursor: pointer;
"></div>
<script>
(function (w, d, s, o, f, js, fjs) {
w['Sales-Nudge-Widget'] = o; w[o] = w[o] || function () { (w[o].q = w[o].q || []).push(arguments) };
js = d.createElement(s), fjs = d.getElementsByTagName(s)[0];
js.id = o; js.src = f; js.async = 1; fjs.parentNode.insertBefore(js, fjs);
}(window, document, 'script', 'wpInstallNudge', 'https://cdn-web-2.ruangguru.com/sales-nudge/6198a95130/dist/widget.js'));
wpInstallNudge('init', { targetElementId: 'wp-install-nudge', clientTarget: 'ruangguru_blog' });
</script>
<script>
// note: disabled due to auto reload page
// var lastScrollTop = 0;
// $(window).scroll(function(event){
// var st = $(this).scrollTop();
// if (st > lastScrollTop){
// $('#wp-install-nudge').slideDown( "slow", function() {});
// } else {
// $('#wp-install-nudge').slideUp( "slow", function() {});
// }
// lastScrollTop = st;
// });
// $(document).ready(function () {
// $("#wp-install-nudge").hide();
// });
$('#wp-install-nudge').click(function(){
Analytics.track("promotion__install_nudge__click",
{
user_email: "ruangguru@blog.com",
cta_link: widgetData.dataConfig.target_link
}).catch((err) => {
console.log("tracker error", err);
});
});
</script>
</footer>
<div id="__next"></div>
<script type="speculationrules">
{"prefetch":[{"source":"document","where":{"and":[{"href_matches":"\/blog\/*"},{"not":{"href_matches":["\/blog\/wp-*.php","\/blog\/wp-admin\/*","\/blog\/wp-content\/uploads\/*","\/blog\/wp-content\/*","\/blog\/wp-content\/plugins\/*","\/blog\/wp-content\/themes\/ruangguru\/*","\/blog\/*\\?(.+)"]}},{"not":{"selector_matches":"a[rel~=\"nofollow\"]"}},{"not":{"selector_matches":".no-prefetch, .no-prefetch a"}}]},"eagerness":"conservative"}]}
</script>
<script>
async function getCtaDetails(ctaIds) {
const endpoint =
" https://gw.ruangguru.com/api/v3/rdb/p_ddgy8hjx5dxsqx/CTA/views/CTA?offset=0&limit=10&where=(ctaId,in," + ctaIds.toString() + ")";
try {
const response = await fetch(endpoint, {
headers: {
"xc-token": "-hG2nBDJzSj0agSgNtcJx97M4F02y54nUMoY6plJ",
},
});
const data = await response.json();
const ctaList = data?.list || [];
const ctaMap = new Map();
for (const ctaData of ctaList) {
ctaMap.set(ctaData.ctaId, ctaData);
}
return ctaMap;
} catch (error) {
console.error("Error fetching CTA details:", error);
return new Map();
}
}
async function replaceCTA() {
const regex = /{{cta\((‘(.*?)’|“(.*?)”)\)}}/g;
const bodyHTML = document.body.innerHTML;
const matches = Array.from(bodyHTML.matchAll(regex));
const ctaIds = matches.map((match) => {
let ctaId = match[2] ?? match[3];
return ctaId;
});
const ctaMap = await getCtaDetails(ctaIds);
const replacedHTML = bodyHTML.replace(regex, (match, _inner, ctaId1, ctaId2) => {
const ctaId = ctaId1 ?? ctaId2;
const ctaData = ctaMap.get(ctaId);
if (ctaData) {
return `<a href="https://cta.ruangguru.com/${ctaData.ctaId}" target="_${ctaData.targetType}" style="position:relative;display:inline-block;">
<img src="${ctaData.imageUrl}" loading="lazy" alt="${ctaData.altText}" />
</a>`;
} else {
return match;
}
});
document.body.innerHTML = replacedHTML;
}
</script>
<meta name="author" content="Ruangguru Tech Team">
<!-- Google Tag Manager (noscript) -->
<noscript><iframe src="https://www.googletagmanager.com/ns.html?id=GTM-PVNLZFT" height="0" width="0" style="display:none;visibility:hidden"></iframe></noscript>
<!-- Tracker -->
<script type="text/javascript" src="https://cdn-web-2.ruangguru.com/landing-page-web/public/staticpages/global-assets/js/cta.js"></script>
<script src="https://unpkg.com/@dotlottie/player-component@latest/dist/dotlottie-player.mjs" type="module"></script>
</body>
</html>