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:

Machine Hours

A measure of the total time that machines are operating, used in cost accounting to allocate machine-related expenses to products.

Allocation Base

A measure or statistic used to allocate costs among different departments, products, or activities within an organization.

Overhead Costs

Expenses not directly tied to production, including rent, utilities, and administrative costs.

Plantwide Overhead Rate

A single overhead absorption rate used throughout an entire plant or facility, applied uniformly to all cost objects irrespective of the department in which they were created.