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) $\quad \mathrm { r } ^ { n + 1 } - 1$
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 sum $S_n = 1 + r + r^2 + \ldots + r^n$ is a geometric series with the first term $a = 1$ and the common ratio $r$. The sum of the first $n+1$ terms of a geometric series is given by the formula $S_n = \frac{a(1 - r^{n+1})}{1 - r}$, where $a = 1$. Substituting $a = 1$ into the formula gives $S_n = \frac{1 - r^{n+1}}{1 - r}$, proving that option A is correct for all natural numbers $n$ and for all real numbers $r \neq 1$.

Select the Right Expression for the Following Sum and Prove nnn

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