# Try to figure out a function f(n) which takes the input n, where n∈N, and outputs the sum ∑i=ni=0kji till the nth value where k∈R & j∈N

Try to figure out a function $f\left(n\right)$ which takes the input n, where $n\in \mathbb{N}$, and outputs the sum $\sum _{i=0}^{i=n}{k}^{ji}$ till the ${n}^{th}$ value where $k\in \mathbb{R}$ & $j\in \mathbb{N}$. For example $\sum _{i=0}^{i=n}{3}^{2i}$
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Jimena Torres
Let $a\in \mathbb{R}$.
$S\left(n\right)=a{k}^{0}+a{k}^{j}+a{k}^{2j}+a{k}^{3j}+\cdots +a{k}^{jn}$
${k}^{j}S\left(n\right)=a{k}^{j}+a{k}^{2j}+a{k}^{3j}+a{k}^{4j}+\cdots +a{k}^{jn+j}$
So,
$S\left(n\right)-{k}^{j}S\left(n\right)=a{k}^{0}-a{k}^{jn+j}$
$S\left(n\right)\left(1-{k}^{j}\right)=a{k}^{0}-a{k}^{jn+j}$
$S\left(n\right)=\frac{a{k}^{0}-a{k}^{jn+j}}{\left(1-{k}^{j}\right)}$
$S\left(n\right)=\frac{a\left(1-{k}^{jn+j}\right)}{\left(1-{k}^{j}\right)}$