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2026-01-01
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<p>Last updated on<strong>August 5, 2025</strong></p>
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<p>Last updated on<strong>August 5, 2025</strong></p>
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<p>In mathematics, sequences and series are fundamental concepts involving ordered lists of numbers and their sums. Sequences can be arithmetic, geometric, or more complex, while series are the sums of sequences. In this topic, we will learn the formulas for various types of sequences and series.</p>
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<p>In mathematics, sequences and series are fundamental concepts involving ordered lists of numbers and their sums. Sequences can be arithmetic, geometric, or more complex, while series are the sums of sequences. In this topic, we will learn the formulas for various types of sequences and series.</p>
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<h2>List of Math Formulas for Sequences and Series</h2>
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<h2>List of Math Formulas for Sequences and Series</h2>
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<p>Sequences and<a>series</a>are key concepts in mathematics. Let’s learn the<a>formulas</a>to calculate the<a>terms</a>and sums of<a>arithmetic</a>and geometric<a>sequences</a>and series.</p>
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<p>Sequences and<a>series</a>are key concepts in mathematics. Let’s learn the<a>formulas</a>to calculate the<a>terms</a>and sums of<a>arithmetic</a>and geometric<a>sequences</a>and series.</p>
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<h2>Math Formula for Arithmetic Sequences</h2>
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<h2>Math Formula for Arithmetic Sequences</h2>
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<p>An<a>arithmetic sequence</a>is a sequence of<a>numbers</a>in which the difference between consecutive terms is<a>constant</a>.</p>
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<p>An<a>arithmetic sequence</a>is a sequence of<a>numbers</a>in which the difference between consecutive terms is<a>constant</a>.</p>
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<p>It is calculated using the formula: [ a_n = a_1 + (n - 1) * d ] where ( a_n ) is the nth term, ( a_1 ) is the first term, \( d \) is the<a>common difference</a>, and \( n \) is the term number.</p>
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<p>It is calculated using the formula: [ a_n = a_1 + (n - 1) * d ] where ( a_n ) is the nth term, ( a_1 ) is the first term, \( d \) is the<a>common difference</a>, and \( n \) is the term number.</p>
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<h2>Math Formula for Arithmetic Series</h2>
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<h2>Math Formula for Arithmetic Series</h2>
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<p>The<a>sum</a>of the first n terms of an arithmetic sequence is called an arithmetic series.</p>
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<p>The<a>sum</a>of the first n terms of an arithmetic sequence is called an arithmetic series.</p>
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<p>The formula is: S_n = n/2 (a_1 + a_n) (or) S_n = n/2 (2a + (n - 1) d) where ( S_n ) is the sum of the first n terms, ( a_1 ) is the first term, ( a_n ) is the nth term, and ( d ) is the common difference.</p>
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<p>The formula is: S_n = n/2 (a_1 + a_n) (or) S_n = n/2 (2a + (n - 1) d) where ( S_n ) is the sum of the first n terms, ( a_1 ) is the first term, ( a_n ) is the nth term, and ( d ) is the common difference.</p>
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<h2>Math Formula for Geometric Sequences</h2>
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<h2>Math Formula for Geometric Sequences</h2>
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<p>A<a>geometric sequence</a>is a sequence of numbers where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the common<a>ratio</a>.</p>
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<p>A<a>geometric sequence</a>is a sequence of numbers where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the common<a>ratio</a>.</p>
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<p>The formula is: [ a_n = a_1 * r(n-1) ] where ( a_n ) is the nth term, ( a_1 ) is the first term, ( r ) is the common ratio, and ( n ) is the term number.</p>
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<p>The formula is: [ a_n = a_1 * r(n-1) ] where ( a_n ) is the nth term, ( a_1 ) is the first term, ( r ) is the common ratio, and ( n ) is the term number.</p>
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<h2>Math Formula for Geometric Series</h2>
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<h2>Math Formula for Geometric Series</h2>
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<p>The sum of the first n terms of a geometric sequence is called a geometric series.</p>
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<p>The sum of the first n terms of a geometric sequence is called a geometric series.</p>
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<p>The formula is: (S_{n} = a_{1}\cdot frac{1-r{n}}{1-r}), where ( S_n ) is the sum of the first n terms, ( a_1) is the first term, and ( r ) is the common ratio.</p>
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<p>The formula is: (S_{n} = a_{1}\cdot frac{1-r{n}}{1-r}), where ( S_n ) is the sum of the first n terms, ( a_1) is the first term, and ( r ) is the common ratio.</p>
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<h2>Tips and Tricks to Memorize Sequences and Series Math Formulas</h2>
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<h2>Tips and Tricks to Memorize Sequences and Series Math Formulas</h2>
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<p>Students often find<a>math</a>formulas tricky and confusing.</p>
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<p>Students often find<a>math</a>formulas tricky and confusing.</p>
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<p>Here are some tips and tricks to master sequences and series formulas:</p>
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<p>Here are some tips and tricks to master sequences and series formulas:</p>
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<ul><li>Use simple mnemonics like "Arithmetic Adds" and "Geometric Grows." </li>
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<ul><li>Use simple mnemonics like "Arithmetic Adds" and "Geometric Grows." </li>
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<li>Connect the use of sequences and series with real-life examples, such as savings plans (series) or patterns (sequences). </li>
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<li>Connect the use of sequences and series with real-life examples, such as savings plans (series) or patterns (sequences). </li>
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<li>Use flashcards to memorize the formulas and rewrite them for quick recall. </li>
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<li>Use flashcards to memorize the formulas and rewrite them for quick recall. </li>
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<li>Create a formula chart for quick reference.</li>
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<li>Create a formula chart for quick reference.</li>
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</ul><h2>Common Mistakes and How to Avoid Them While Using Sequences and Series Math Formulas</h2>
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</ul><h2>Common Mistakes and How to Avoid Them While Using Sequences and Series Math Formulas</h2>
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<p>Students often make errors when calculating sequences and series. Here are some mistakes and ways to avoid them to master these concepts.</p>
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<p>Students often make errors when calculating sequences and series. Here are some mistakes and ways to avoid them to master these concepts.</p>
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<h3>Problem 1</h3>
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<h3>Problem 1</h3>
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<p>Find the 10th term of the arithmetic sequence where the first term is 3 and the common difference is 5.</p>
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<p>Find the 10th term of the arithmetic sequence where the first term is 3 and the common difference is 5.</p>
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<p>Okay, lets begin</p>
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<p>Okay, lets begin</p>
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<p>The 10th term is 48.</p>
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<p>The 10th term is 48.</p>
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<h3>Explanation</h3>
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<h3>Explanation</h3>
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<p>Using the formula ( a_n = a_1 + (n - 1) cdot d ):</p>
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<p>Using the formula ( a_n = a_1 + (n - 1) cdot d ):</p>
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<p> a_{10} = 3 + (10 - 1) cdot 5</p>
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<p> a_{10} = 3 + (10 - 1) cdot 5</p>
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<p>= 3 + 45</p>
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<p>= 3 + 45</p>
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<p>= 48 </p>
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<p>= 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>Calculate the sum of the first 8 terms of the arithmetic sequence where the first term is 4 and the common difference is 3.</p>
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<p>Calculate the sum of the first 8 terms of the arithmetic sequence where the first term is 4 and the common difference is 3.</p>
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<p>Okay, lets begin</p>
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<p>Okay, lets begin</p>
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<p>The sum is 100.</p>
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<p>The sum is 100.</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 = frac{n}{2} cdot (2a_1 + (n - 1) cdot d) ):</p>
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<p>Using the formula ( S_n = frac{n}{2} cdot (2a_1 + (n - 1) cdot d) ):</p>
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<p>S_8 = frac{8}{2} cdot (2 cdot 4 + (8 - 1) cdot 3)</p>
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<p>S_8 = frac{8}{2} cdot (2 cdot 4 + (8 - 1) cdot 3)</p>
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<p>= 4 cdot (8 + 21)</p>
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<p>= 4 cdot (8 + 21)</p>
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<p>= 4 cdot 29</p>
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<p>= 4 cdot 29</p>
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<p>= 116 </p>
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<p>= 116 </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 6th term of the geometric sequence where the first term is 2 and the common ratio is 3.</p>
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<p>Find the 6th term of the geometric sequence where the first term is 2 and the common ratio is 3.</p>
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<p>Okay, lets begin</p>
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<p>Okay, lets begin</p>
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<p>The 6th term is 486.</p>
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<p>The 6th term is 486.</p>
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<h3>Explanation</h3>
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<h3>Explanation</h3>
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<p>Using the formula ( a_n = a_1 cdot r{(n-1)} ):</p>
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<p>Using the formula ( a_n = a_1 cdot r{(n-1)} ):</p>
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<p>a_6 = 2 cdot 3{(6-1)}</p>
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<p>a_6 = 2 cdot 3{(6-1)}</p>
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<p>= 2 cdot 243</p>
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<p>= 2 cdot 243</p>
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<p>= 486 </p>
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<p>= 486 </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>Calculate the sum of the first 5 terms of the geometric sequence where the first term is 1 and the common ratio is 2.</p>
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<p>Calculate the sum of the first 5 terms of the geometric sequence where the first term is 1 and the common ratio is 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>The sum is 31.</p>
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<p>The sum is 31.</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 cdot frac{1 - rn}{1 - r} ):</p>
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<p>Using the formula ( S_n = a_1 cdot frac{1 - rn}{1 - r} ):</p>
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<p>S_5 = 1 cdot frac{1 - 25}{1 - 2}</p>
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<p>S_5 = 1 cdot frac{1 - 25}{1 - 2}</p>
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<p>= frac{1 - 32}{-1}</p>
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<p>= frac{1 - 32}{-1}</p>
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<p>= 31</p>
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<p>= 31</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>What is the 7th term of the arithmetic sequence where the first term is 10 and the common difference is 4?</p>
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<p>What is the 7th term of the arithmetic sequence where the first term is 10 and the common difference is 4?</p>
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<p>Okay, lets begin</p>
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<p>Okay, lets begin</p>
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<p>The 7th term is 34.</p>
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<p>The 7th term is 34.</p>
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<h3>Explanation</h3>
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<h3>Explanation</h3>
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<p>Using the formula ( a_n = a_1 + (n - 1) cdot d ): </p>
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<p>Using the formula ( a_n = a_1 + (n - 1) cdot d ): </p>
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<p>a_7 = 10 + (7 - 1) cdot 4</p>
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<p>a_7 = 10 + (7 - 1) cdot 4</p>
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<p>= 10 + 24</p>
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<p>= 10 + 24</p>
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<p>= 34 .</p>
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<p>= 34 .</p>
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<p>Well explained 👍</p>
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<p>Well explained 👍</p>
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<h2>FAQs on Sequences and Series Math Formulas</h2>
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<h2>FAQs on Sequences and Series Math Formulas</h2>
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<h3>1.What is the formula for an arithmetic sequence?</h3>
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<h3>1.What is the formula for an arithmetic sequence?</h3>
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<p>The formula to find the nth term of an arithmetic sequence is: \( a_n = a_1 + (n - 1) \cdot d \).</p>
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<p>The formula to find the nth term of an arithmetic sequence is: \( a_n = a_1 + (n - 1) \cdot d \).</p>
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<h3>2.What is the formula for the sum of an arithmetic series?</h3>
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<h3>2.What is the formula for the sum of an arithmetic series?</h3>
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<p>The formula for the sum of the first n terms of an arithmetic series is: \( S_n = \frac{n}{2} \cdot (a_1 + a_n) \).</p>
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<p>The formula for the sum of the first n terms of an arithmetic series is: \( S_n = \frac{n}{2} \cdot (a_1 + a_n) \).</p>
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<h3>3.What is the formula for a geometric sequence?</h3>
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<h3>3.What is the formula for a geometric sequence?</h3>
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<p>The formula to find the nth term of a geometric sequence is: \( a_n = a_1 \cdot r^{(n-1)} \).</p>
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<p>The formula to find the nth term of a geometric sequence is: \( a_n = a_1 \cdot r^{(n-1)} \).</p>
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<h3>4.What is the formula for the sum of a geometric series?</h3>
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<h3>4.What is the formula for the sum of a geometric series?</h3>
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<p>The formula for the sum of the first n terms of a geometric series is: \( S_n = a_1 \cdot \frac{1 - r^n}{1 - r} \) for \( r \neq 1 \).</p>
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<p>The formula for the sum of the first n terms of a geometric series is: \( S_n = a_1 \cdot \frac{1 - r^n}{1 - r} \) for \( r \neq 1 \).</p>
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<h3>5.How do I identify if a sequence is arithmetic or geometric?</h3>
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<h3>5.How do I identify if a sequence is arithmetic or geometric?</h3>
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<p>A sequence is arithmetic if the difference between consecutive terms is constant (common difference). It is geometric if the ratio between consecutive terms is constant (common ratio).</p>
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<p>A sequence is arithmetic if the difference between consecutive terms is constant (common difference). It is geometric if the ratio between consecutive terms is constant (common ratio).</p>
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<h2>Glossary for Sequences and Series Math Formulas</h2>
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<h2>Glossary for Sequences and Series Math Formulas</h2>
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<ul><li><strong>Arithmetic Sequence:</strong>A sequence in which each term is derived by adding a constant to the previous term.</li>
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<ul><li><strong>Arithmetic Sequence:</strong>A sequence in which each term is derived by adding a constant to the previous term.</li>
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</ul><ul><li><strong>Geometric Sequence:</strong>A sequence in which each term is derived by multiplying the previous term by a constant.</li>
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</ul><ul><li><strong>Geometric Sequence:</strong>A sequence in which each term is derived by multiplying the previous term by a constant.</li>
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</ul><ul><li><strong>Arithmetic Series:</strong>The sum of the terms in an arithmetic sequence.</li>
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</ul><ul><li><strong>Arithmetic Series:</strong>The sum of the terms in an arithmetic sequence.</li>
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</ul><ul><li><strong>Geometric Series:</strong>The sum of the terms in a geometric sequence.</li>
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</ul><ul><li><strong>Geometric Series:</strong>The sum of the terms in a geometric sequence.</li>
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</ul><ul><li><strong>Common Ratio:</strong>The fixed number that each term in a geometric sequence is multiplied by to get the next term.</li>
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</ul><ul><li><strong>Common Ratio:</strong>The fixed number that each term in a geometric sequence is multiplied by to get the next term.</li>
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</ul><h2>Jaskaran Singh Saluja</h2>
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</ul><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>