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

question 155

Multiple Choice

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) \(\frac { 1 - p ^ { n } } { 1 - p }\)

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Definitions:

Market Share

The portion of a market controlled by a particular company, product, or brand, typically expressed as a percentage of total sales in the industry.