Select the right expression for the following sum and prove your statement is true for all natural numbers $n$, where $r$ is a real number, $r \neq 1$.
$S _ { n } : 1 + r + r ^ { 2 } + \ldots + r ^ { n }$
A) $\quad \frac { 1 - r ^ { n + 1 } } { 1 - r }$
B) $\quad r ^ { n + 1 } - 1$
C) $\quad \frac { r ^ { n + 1 } } { r - 1 }$
D) $\quad \frac { 1 - r ^ { n } } { 1 - r }$
E) $\quad \frac { 1 - r ^ { n } } { 1 - r ^ { n - 1 } }$

Question

Quizplus · Accepted Answer

The correct expression for the sum $S_n: 1 + r + r^2 + \ldots + r^n$ is given by $ \frac { 1 - r ^ { n + 1 } } { 1 - r }$. This can be proven by considering the sum $S_n$ and multiplying it by $r$, then subtracting the result from the original sum to eliminate all terms except the first and the last. This process leads to a simple equation that can be solved for $S_n$, yielding the formula $ \frac { 1 - r ^ { n + 1 } } { 1 - r }$ as the solution. This formula is valid for all natural numbers $n$ and for any real number $r$ different from 1.

Select the Right Expression for the Following Sum and Prove nnn

Select the Right Expression for the Following Sum and Prove $n$