How do you find the sum of finite geometric series?

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How do you find the sum of finite geometric series?

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Answer 1 · 2015-07-09T03:18:50

$a + a r + a r^{2} + \dots + a r^{n} = \frac{a (r^{n + 1} - 1)}{r - 1}$ $a+ar+ar^2+\cdots+ar^n=\frac{a(r^{n+1}-1)}{r-1}$ if $r \neq 1$ $r!=1$ and $a + a + a + \dots + a = (n + 1) a$ $a+a+a+\cdots+a=(n+1)a$ when there are $n + 1$ $n+1$ terms (and $r = 1$ $r=1$ ).

Explanation:

Consider a finite geometric series with $n + 1$ $n+1$ terms $a + a r + a r^{2} + \dots + a r^{n}$ $a+ar+ar^2+\cdots+ar^n$ . Call this sum $S_{n}$ $S_{n}$ . Note that $r S_{n} = a r + a r^{2} + a r^{3} + \dots + a r^{n + 1}$ $rS_{n}=ar+ar^2+ar^3+\cdots+ar^{n+1}$ so that $r S_{n} - S_{n} = a r^{n + 1} - a$ $rS_{n}-S_{n}=ar^{n+1}-a$ .

Solving this equation for $S_{n}$ $S_{n}$ gives $S_{n} = \frac{a (r^{n + 1} - 1)}{r - 1}$ $S_{n}=\frac{a(r^{n+1}-1)}{r-1}$ when $r \neq 1$ $r!=1$ . The number $a$ $a$ is often referred to as the "first term" and the number $r$ $r$ is often referred to as the "common ratio". The number $n$ $n$ is the highest power of $r$ $r$ in the sum and the sum itself has $n + 1$ $n+1$ terms.

If $r = 1$ $r=1$ , then $S_{n} = a + a + a + \dots + a = (n + 1) a$ $S_{n}=a+a+a+\cdots+a=(n+1)a$ .

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How do you find the sum of finite geometric series?

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