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

Question

Quizplus · Accepted Answer

The sum $S_n: 1 + p + p^2 + \ldots + p^n$ is a geometric series with the first term $a = 1$ and common ratio $r = p$. The sum of the first $n+1$ terms of a geometric series is given by $S_n = \frac{a(1 - r^{n+1})}{1 - r}$, where $n+1$ is the total number of terms. Substituting $a = 1$ and $r = p$, we get $S_n = \frac{1(1 - p^{n+1})}{1 - p} = \frac{1 - p^{n+1}}{1 - p}$, which matches option B. This formula is valid for all natural numbers $n$ and for any real number $p$, provided $p \neq 1$, to avoid division by zero.

Select the Right Expression for the Following Sum and Prove nnn

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