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

question 103

Multiple Choice

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) \(\quad \frac { 1 - r ^ { n } } { 1 - r ^ { n - 1 } }\)

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Definitions:

Genes

Units of heredity made up of DNA, responsible for guiding the development and functioning of organisms.

Nucleotides

Organic molecules that serve as the monomers, or subunits, of nucleic acids like DNA and RNA, composed of a phosphate group, a sugar, and a nitrogenous base.

Venus Flytrap

A carnivorous plant that captures and digests insects and arthropods by snapping shut its modified leaves.

Stimulus

An external factor or event that triggers a specific reaction in an organism or system.