What is the formula for the sum of a geometric sequence?

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Answer 1 · 2015-09-02T10:06:37

The formula is: $S_n=(a_1(1-q^n))/(1-q)$

To proove this formula you have first to write the sum of a geometric sequence:

$S_n=a_1+a_1*q+a_1*q^2+...a_1*q^(n-1)$

$S_n=a_1*(1+q+q^2+...q^n)$

We can multiply the last equality by $(1-q)$
We get:

$(1-q)*S_n=a_1*(1-q)*(1+q+q^2+...q^n)$

The right hand side can be written as $a_1*(1^n-q^n)$ or
$a_1*(1-q^n)$ . So finally we get:

$(1-q)*S_n=a_1*(1-q^n)$

Supposing $q!=1$ , we can divide both sides by $1-q$ and get:

$S_n=(a_1(1-q^n))/(1-q)$

That concludes the proof.