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2026-01-01
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<p>Last updated on<strong>October 18, 2025</strong></p>
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<p>Last updated on<strong>October 18, 2025</strong></p>
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<p>ChatGPT said: A Geometric Progression (GP) is a sequence where each term is obtained by multiplying the previous term by a fixed common ratio. The previous term can be found by dividing by this ratio. For example, 3, 6, 12, 24,… is a GP with a ratio of 2. GPs can have finite or infinite terms, and this article covers their meaning, formulas, and types.</p>
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<p>ChatGPT said: A Geometric Progression (GP) is a sequence where each term is obtained by multiplying the previous term by a fixed common ratio. The previous term can be found by dividing by this ratio. For example, 3, 6, 12, 24,… is a GP with a ratio of 2. GPs can have finite or infinite terms, and this article covers their meaning, formulas, and types.</p>
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<h2>What is a Geometric Progression?</h2>
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<h2>What is a Geometric Progression?</h2>
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<p>What Is Algebra? 🧮 | Simple Explanation with 🎯 Cool Examples for Kids | ✨BrightCHAMPS Math</p>
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<p>What Is Algebra? 🧮 | Simple Explanation with 🎯 Cool Examples for Kids | ✨BrightCHAMPS Math</p>
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<p>▶</p>
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<p>The<a>sequence</a>in which each<a>term</a>is obtained by multiplying the previous term by a fixed<a>number</a>(common<a>ratio</a>) is known as a geometric<a>progression</a>. It is usually expressed as: \(a, ar, ar^2, ar^3…\), where ‘a’ represents the first term and ‘r’ represents the common<a>ratio</a>. The common ratio can be positive or negative. Any term in a GP can be determined using the first term and the common ratio.</p>
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<p>The<a>sequence</a>in which each<a>term</a>is obtained by multiplying the previous term by a fixed<a>number</a>(common<a>ratio</a>) is known as a geometric<a>progression</a>. It is usually expressed as: \(a, ar, ar^2, ar^3…\), where ‘a’ represents the first term and ‘r’ represents the common<a>ratio</a>. The common ratio can be positive or negative. Any term in a GP can be determined using the first term and the common ratio.</p>
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<h2>What are the Types of Geometric Progression?</h2>
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<h2>What are the Types of Geometric Progression?</h2>
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<p>Geometric progressions are mainly classified into two types based on their length.</p>
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<p>Geometric progressions are mainly classified into two types based on their length.</p>
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<p>The different types of geometric progressions are:</p>
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<p>The different types of geometric progressions are:</p>
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<ul><li>Finite geometric progression </li>
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<ul><li>Finite geometric progression </li>
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<li>Infinite geometric progression</li>
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<li>Infinite geometric progression</li>
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</ul><p>We will now learn about each type in detail:</p>
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</ul><p>We will now learn about each type in detail:</p>
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<p><strong>Finite geometric progression: </strong>A finite geometric progression has a limited number<a>of terms</a>, and the last term is known. For example: \(\frac{1}{2}\), \(\frac{1}{4}\), \(\frac{1}{8}\), \(\frac{1}{16}\), …, \(\frac{1}{32768}\) is a finite<a>geometric</a>progression. Here, \(\frac{1}{32768}\) is the last term.</p>
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<p><strong>Finite geometric progression: </strong>A finite geometric progression has a limited number<a>of terms</a>, and the last term is known. For example: \(\frac{1}{2}\), \(\frac{1}{4}\), \(\frac{1}{8}\), \(\frac{1}{16}\), …, \(\frac{1}{32768}\) is a finite<a>geometric</a>progression. Here, \(\frac{1}{32768}\) is the last term.</p>
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<p><strong>Infinite geometric progression: </strong>An infinite geometric progression has an endless number of terms. Since there is no fixed number of terms, the last term cannot be specified. For example, the infinite<a>series</a>3, -6, 12, -24, … does not have a definite end term.</p>
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<p><strong>Infinite geometric progression: </strong>An infinite geometric progression has an endless number of terms. Since there is no fixed number of terms, the last term cannot be specified. For example, the infinite<a>series</a>3, -6, 12, -24, … does not have a definite end term.</p>
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<h2>GP vs AP</h2>
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<h2>GP vs AP</h2>
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<p>To help you identify the sequence effectively, we will now look at the key differences between GP and AP.</p>
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<p>To help you identify the sequence effectively, we will now look at the key differences between GP and AP.</p>
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<strong>Geometric Progression (GP)</strong><strong>Arithmetic Progression (AP)</strong>Each term is obtained by multiplying the previous term by a fixed common ratio 𝑟. Each term is obtained by adding a fixed<a>common difference</a> 𝑑 to the previous term. No common difference between the terms. There is no fixed ratio between the terms For example: 2, 4, 8, 16,...(r = 2) For example: 3, 6, 9, 12,...(d = 3) Such series can converge or diverge depending on r. The series is always divergent unless the common difference is zero. Formula for n-th term is \(a_n = a_1 \cdot r^{\,n-1} \). Formula for n-th term is \(a_n = a_1 + (n-1)d \).<h3>Explore Our Programs</h3>
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<strong>Geometric Progression (GP)</strong><strong>Arithmetic Progression (AP)</strong>Each term is obtained by multiplying the previous term by a fixed common ratio 𝑟. Each term is obtained by adding a fixed<a>common difference</a> 𝑑 to the previous term. No common difference between the terms. There is no fixed ratio between the terms For example: 2, 4, 8, 16,...(r = 2) For example: 3, 6, 9, 12,...(d = 3) Such series can converge or diverge depending on r. The series is always divergent unless the common difference is zero. Formula for n-th term is \(a_n = a_1 \cdot r^{\,n-1} \). Formula for n-th term is \(a_n = a_1 + (n-1)d \).<h3>Explore Our Programs</h3>
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<h2>What are the Properties of GP?</h2>
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<h2>What are the Properties of GP?</h2>
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<p>Understanding the unique features of a progression helps us identify it more easily. Here are a few properties that geometric progressions (GP) follow. </p>
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<p>Understanding the unique features of a progression helps us identify it more easily. Here are a few properties that geometric progressions (GP) follow. </p>
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<ul><li>The<a>square</a>of any term in a GP is equal to the<a>product</a>of the terms that are directly adjacent to it: \(a_k² = a_{k-1} × a_{k+1}\)</li>
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<ul><li>The<a>square</a>of any term in a GP is equal to the<a>product</a>of the terms that are directly adjacent to it: \(a_k² = a_{k-1} × a_{k+1}\)</li>
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</ul><ul><li>In a finite geometric progression, terms that are equally spaced from the beginning and the end have the same product: \(a_1 × a_n = a_2 × a_{ n-1} =…= a_k × a_{n-k+1}\)</li>
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</ul><ul><li>In a finite geometric progression, terms that are equally spaced from the beginning and the end have the same product: \(a_1 × a_n = a_2 × a_{ n-1} =…= a_k × a_{n-k+1}\)</li>
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</ul><ul><li>Multiplying or<a>dividing</a>a GP by a non-zero<a>constant</a>, the new sequence remains a GP with the same common ratio.</li>
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</ul><ul><li>Multiplying or<a>dividing</a>a GP by a non-zero<a>constant</a>, the new sequence remains a GP with the same common ratio.</li>
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</ul><ul><li>The reciprocal of each term in a GP results in another GP with a new common ratio equal to 1/r (r = original common ratio).</li>
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</ul><ul><li>The reciprocal of each term in a GP results in another GP with a new common ratio equal to 1/r (r = original common ratio).</li>
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</ul><ul><li>A GP remains a GP even if each of its terms is raised to the same<a>power</a>. Example: GP: a, ar, ar2,...,<p>Raise each term to the power k: \(a^k, (ar)^k, (ar²)^k, …\) which is still a GP.</p>
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</ul><ul><li>A GP remains a GP even if each of its terms is raised to the same<a>power</a>. Example: GP: a, ar, ar2,...,<p>Raise each term to the power k: \(a^k, (ar)^k, (ar²)^k, …\) which is still a GP.</p>
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</li>
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</li>
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</ul><h2>What is the Formula for GP?</h2>
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</ul><h2>What is the Formula for GP?</h2>
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<p>In a GP, the<a>sum</a>of the terms can be calculated using the following<a>formulas</a>:</p>
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<p>In a GP, the<a>sum</a>of the terms can be calculated using the following<a>formulas</a>:</p>
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<p>For a \(GP: a, ar, ar^2, ar^3\), … </p>
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<p>For a \(GP: a, ar, ar^2, ar^3\), … </p>
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<ul><li><strong>nth term:</strong>\(a_n = a × r^{n-1} \space \text{or} \space a_n = r × a_{n-1}\) </li>
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<ul><li><strong>nth term:</strong>\(a_n = a × r^{n-1} \space \text{or} \space a_n = r × a_{n-1}\) </li>
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</ul><ul><li><strong>Sum of the first n terms:</strong>\(S_n = a(1 - r^n)/(1 - r), for \space r ≠ 1\) \(S_n = n × a\) for r = 1 </li>
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</ul><ul><li><strong>Sum of the first n terms:</strong>\(S_n = a(1 - r^n)/(1 - r), for \space r ≠ 1\) \(S_n = n × a\) for r = 1 </li>
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</ul><ul><li><strong>Sum of infinite terms: </strong>\(S_∞ = a / (1 - r)\), when |r| < 1</li>
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</ul><ul><li><strong>Sum of infinite terms: </strong>\(S_∞ = a / (1 - r)\), when |r| < 1</li>
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</ul><p>The sum does not exist when \(|r| ≥ 1\).</p>
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</ul><p>The sum does not exist when \(|r| ≥ 1\).</p>
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<h2>Tips and Tricks to Master Geometric Progression</h2>
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<h2>Tips and Tricks to Master Geometric Progression</h2>
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<p>Learn how to quickly identify, analyze, and apply geometric progressions in problems and real-life scenarios.</p>
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<p>Learn how to quickly identify, analyze, and apply geometric progressions in problems and real-life scenarios.</p>
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<ul><li>Identify the common ratio to determine growth or decay. </li>
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<ul><li>Identify the common ratio to determine growth or decay. </li>
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<li>Apply the n-th term formula. </li>
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<li>Apply the n-th term formula. </li>
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<li>Memorize finite and infinite sum formulas. </li>
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<li>Memorize finite and infinite sum formulas. </li>
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<li>Determine if the series converges or diverges. </li>
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<li>Determine if the series converges or diverges. </li>
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<li>Solve real-life problems using GP concepts.</li>
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<li>Solve real-life problems using GP concepts.</li>
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</ul><h2>Common Mistakes and How to Avoid Them in Geometric Progression</h2>
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</ul><h2>Common Mistakes and How to Avoid Them in Geometric Progression</h2>
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<p>Geometric progression is a simple mathematical concept, but many students struggle with its problems. Here are a few common mistakes and tips to avoid them:</p>
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<p>Geometric progression is a simple mathematical concept, but many students struggle with its problems. Here are a few common mistakes and tips to avoid them:</p>
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<h2>Real-Life Applications of Geometric Progression</h2>
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<h2>Real-Life Applications of Geometric Progression</h2>
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<p>Geometric progression has a vital role in various real-life situations. Let's explore how this concept applies in real-life scenarios.</p>
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<p>Geometric progression has a vital role in various real-life situations. Let's explore how this concept applies in real-life scenarios.</p>
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<ul><li><strong>Compound Interest in Finance - </strong>The amount of<a>money</a>grows exponentially when interest is compounded periodically, forming a GP. </li>
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<ul><li><strong>Compound Interest in Finance - </strong>The amount of<a>money</a>grows exponentially when interest is compounded periodically, forming a GP. </li>
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<li><strong>Population Growth - </strong>Populations that grow by a fixed<a>percentage</a>over time follow a geometric progression. </li>
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<li><strong>Population Growth - </strong>Populations that grow by a fixed<a>percentage</a>over time follow a geometric progression. </li>
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<li><strong>Depreciation of Assets - </strong>The value of machinery or vehicles often decreases by a fixed ratio annually, modeled using GP. </li>
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<li><strong>Depreciation of Assets - </strong>The value of machinery or vehicles often decreases by a fixed ratio annually, modeled using GP. </li>
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<li><strong>Radioactive Decay - </strong>The remaining quantity of a radioactive substance decreases by a fixed<a>fraction</a>over equal time intervals. </li>
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<li><strong>Radioactive Decay - </strong>The remaining quantity of a radioactive substance decreases by a fixed<a>fraction</a>over equal time intervals. </li>
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<li><strong>Computer Science & Algorithms - </strong>Problems like binary search or tree structures involve<a>exponential growth</a>or halving, which are modeled by geometric progressions.</li>
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<li><strong>Computer Science & Algorithms - </strong>Problems like binary search or tree structures involve<a>exponential growth</a>or halving, which are modeled by geometric progressions.</li>
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</ul><h3>Problem 1</h3>
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</ul><h3>Problem 1</h3>
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<p>Find the 5ᵗʰ term of a GP Given: First term (a) = 3 Common ratio (r) = 2</p>
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<p>Find the 5ᵗʰ term of a GP Given: First term (a) = 3 Common ratio (r) = 2</p>
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<p>Okay, lets begin</p>
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<p>Okay, lets begin</p>
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<p>a5 = 48</p>
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<p>a5 = 48</p>
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<h3>Explanation</h3>
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<h3>Explanation</h3>
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<p>First, apply the formula for the nᵗʰ term: \(a_n = a \cdot r^{\,n-1} \)</p>
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<p>First, apply the formula for the nᵗʰ term: \(a_n = a \cdot r^{\,n-1} \)</p>
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<p>Substituting the values into the formula: \(a_5 = 3 \times 2^{\,5-1} = 3 \times 2^4 \)</p>
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<p>Substituting the values into the formula: \(a_5 = 3 \times 2^{\,5-1} = 3 \times 2^4 \)</p>
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<p>Here, we get: a5 = 3 × 16 = 48</p>
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<p>Here, we get: a5 = 3 × 16 = 48</p>
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<p>Well explained 👍</p>
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<p>Well explained 👍</p>
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<h3>Problem 2</h3>
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<h3>Problem 2</h3>
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<p>Find the sum to infinity of a GP Given: a = 8, r = 1/2</p>
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<p>Find the sum to infinity of a GP Given: a = 8, r = 1/2</p>
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<p>Okay, lets begin</p>
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<p>Okay, lets begin</p>
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<p>S∞ = 16</p>
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<p>S∞ = 16</p>
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<h3>Explanation</h3>
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<h3>Explanation</h3>
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<p>Let’s first check if |r| < 1</p>
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<p>Let’s first check if |r| < 1</p>
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<p>It holds true for the infinite sum since |1/2| < 1.</p>
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<p>It holds true for the infinite sum since |1/2| < 1.</p>
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<p>Using the formula: S∞ = a / (1 - r)</p>
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<p>Using the formula: S∞ = a / (1 - r)</p>
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<p>Substituting the values into the formula: S∞ = 8 / (1 - 1/2) = 8 / (1/2)</p>
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<p>Substituting the values into the formula: S∞ = 8 / (1 - 1/2) = 8 / (1/2)</p>
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<p>So, S∞ = 8 × 2 = 16</p>
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<p>So, S∞ = 8 × 2 = 16</p>
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<p>Well explained 👍</p>
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<p>Well explained 👍</p>
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<h3>Problem 3</h3>
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<h3>Problem 3</h3>
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<p>Find the sum of the first 6 terms of a GP Given: a = 5, r = 3, n = 6</p>
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<p>Find the sum of the first 6 terms of a GP Given: a = 5, r = 3, n = 6</p>
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<p>Okay, lets begin</p>
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<p>Okay, lets begin</p>
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<p>S6 = 1820</p>
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<p>S6 = 1820</p>
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<h3>Explanation</h3>
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<h3>Explanation</h3>
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<p>Here, we use the formula for the sum of the first n terms</p>
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<p>Here, we use the formula for the sum of the first n terms</p>
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<p>\(S_n = a(1 - r^n)/(1 - r)\)</p>
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<p>\(S_n = a(1 - r^n)/(1 - r)\)</p>
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<p>Let’s substitute the values: S6 = 5(36 - 1) / (3 - 1)</p>
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<p>Let’s substitute the values: S6 = 5(36 - 1) / (3 - 1)</p>
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<p>We now calculate powers and simplify: 36 = 729 S6 = 5 (729 - 1) / 2 = (5 × 728) / 2</p>
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<p>We now calculate powers and simplify: 36 = 729 S6 = 5 (729 - 1) / 2 = (5 × 728) / 2</p>
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<p>So, S6 = 3640 / 2 = 1820</p>
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<p>So, S6 = 3640 / 2 = 1820</p>
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<p>Well explained 👍</p>
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<p>Well explained 👍</p>
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<h3>Problem 4</h3>
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<h3>Problem 4</h3>
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<p>Find the 8ᵗʰ term of the GP 5, 10, 20, 40,... Given: a = 5, r = 2, n = 8</p>
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<p>Find the 8ᵗʰ term of the GP 5, 10, 20, 40,... Given: a = 5, r = 2, n = 8</p>
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<p>Okay, lets begin</p>
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<p>Okay, lets begin</p>
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<p>a8 = 640</p>
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<p>a8 = 640</p>
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<h3>Explanation</h3>
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<h3>Explanation</h3>
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<p>Here, we apply the formula for the nᵗʰ term:</p>
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<p>Here, we apply the formula for the nᵗʰ term:</p>
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<p>\(a_n = a \times r^{\,n-1} \)</p>
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<p>\(a_n = a \times r^{\,n-1} \)</p>
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<p>Substituting the values into the formula: a8 = 5 × 28 - 1 = 5 × 27</p>
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<p>Substituting the values into the formula: a8 = 5 × 28 - 1 = 5 × 27</p>
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<p>So, a8 = 5 × 128 = 640</p>
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<p>So, a8 = 5 × 128 = 640</p>
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<p>Well explained 👍</p>
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<p>Well explained 👍</p>
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<h3>Problem 5</h3>
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<h3>Problem 5</h3>
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<p>Find how many terms of the GP 3, 6, 12, 24,... are needed to make the sum 93 Given: a = 3, r = 2, Sₙ = 93</p>
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<p>Find how many terms of the GP 3, 6, 12, 24,... are needed to make the sum 93 Given: a = 3, r = 2, Sₙ = 93</p>
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<p>Okay, lets begin</p>
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<p>Okay, lets begin</p>
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<p>n = 5</p>
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<p>n = 5</p>
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<h3>Explanation</h3>
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<h3>Explanation</h3>
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<p>Using the formula: \(S_n = a(1 - r^n)/(1 - r)\)</p>
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<p>Using the formula: \(S_n = a(1 - r^n)/(1 - r)\)</p>
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<p>Substituting the given values: 93 = 3(2n - 1) / (2 - 1)</p>
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<p>Substituting the given values: 93 = 3(2n - 1) / (2 - 1)</p>
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<p>Now, simplify to get the result:</p>
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<p>Now, simplify to get the result:</p>
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<p>93 = 3(2n - 1)</p>
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<p>93 = 3(2n - 1)</p>
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<p>93 ÷ 3 = 2n - 1</p>
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<p>93 ÷ 3 = 2n - 1</p>
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<p>31 = 2n - 1</p>
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<p>31 = 2n - 1</p>
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<p>2n = 31 + 1</p>
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<p>2n = 31 + 1</p>
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<p>2n = 32 </p>
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<p>2n = 32 </p>
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<p>Here, n is the exponent to which 2 needs to be raised to obtain 32.</p>
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<p>Here, n is the exponent to which 2 needs to be raised to obtain 32.</p>
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<p>Since 25 = 32 → n = 5</p>
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<p>Since 25 = 32 → n = 5</p>
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<p>Well explained 👍</p>
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<p>Well explained 👍</p>
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<h2>FAQs on Geometric Progression</h2>
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<h2>FAQs on Geometric Progression</h2>
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<h3>1.What is meant by the term GP?</h3>
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<h3>1.What is meant by the term GP?</h3>
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<p>GP stands for Geometric Progression. In a GP, each term is the product of the term before it and a constant value called the common ratio.</p>
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<p>GP stands for Geometric Progression. In a GP, each term is the product of the term before it and a constant value called the common ratio.</p>
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<h3>2.Give the formula for the nth term of a GP.</h3>
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<h3>2.Give the formula for the nth term of a GP.</h3>
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<p>The formula for the nth term is: an = a1 × r (n -1)</p>
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<p>The formula for the nth term is: an = a1 × r (n -1)</p>
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<p>Here: </p>
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<p>Here: </p>
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<p>an = nth term</p>
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<p>an = nth term</p>
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<p>a1 = first term</p>
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<p>a1 = first term</p>
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<p>r = common ratio</p>
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<p>r = common ratio</p>
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<p>n = term number</p>
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<p>n = term number</p>
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<h3>3.What do you mean by a common ratio?</h3>
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<h3>3.What do you mean by a common ratio?</h3>
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<p>The constant<a>factor</a>by which each term of the progression is multiplied to obtain the subsequent term is known as the common ratio (r). For example: GP: 2, 4, 8, 16; r = 2.</p>
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<p>The constant<a>factor</a>by which each term of the progression is multiplied to obtain the subsequent term is known as the common ratio (r). For example: GP: 2, 4, 8, 16; r = 2.</p>
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<h3>4.Can the common ratio be 1?</h3>
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<h3>4.Can the common ratio be 1?</h3>
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<p>Yes, the common ratio can be 1. When r = 1, each term in the geometric progression will be the same as the first term. Example: 5, 5, 5, 5, 5…</p>
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<p>Yes, the common ratio can be 1. When r = 1, each term in the geometric progression will be the same as the first term. Example: 5, 5, 5, 5, 5…</p>
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<h3>5.Is it possible for a geometric progression's common ratio to be negative?</h3>
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<h3>5.Is it possible for a geometric progression's common ratio to be negative?</h3>
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<p>Yes, a common ratio can be either positive or negative. If the common ratio is negative, then the terms in the progression will alternate between positive and negative values. For example: 2, -4, 8, -16, 32,…</p>
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<p>Yes, a common ratio can be either positive or negative. If the common ratio is negative, then the terms in the progression will alternate between positive and negative values. For example: 2, -4, 8, -16, 32,…</p>
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<h2>Jaskaran Singh Saluja</h2>
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<h2>Jaskaran Singh Saluja</h2>
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<h3>About the Author</h3>
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<h3>About the Author</h3>
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<p>Jaskaran Singh Saluja is a math wizard with nearly three years of experience as a math teacher. His expertise is in algebra, so he can make algebra classes interesting by turning tricky equations into simple puzzles.</p>
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<p>Jaskaran Singh Saluja is a math wizard with nearly three years of experience as a math teacher. His expertise is in algebra, so he can make algebra classes interesting by turning tricky equations into simple puzzles.</p>
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<h3>Fun Fact</h3>
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<h3>Fun Fact</h3>
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<p>: He loves to play the quiz with kids through algebra to make kids love it.</p>
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<p>: He loves to play the quiz with kids through algebra to make kids love it.</p>