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

question 134

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) \(\quad \mathrm { r } ^ { n + 1 } - 1\)
• C) \(\quad \frac { r ^ { n + 1 } } { r - 1 }\)
• D) \(\quad \frac { 1 - r ^ { n } } { 1 - r }\)
• E) \(\quad \frac { 1 - r ^ { n } } { 1 - r ^ { n - 1 } }\)

Analyze factors influencing the cost of insurance and the role of public transportation systems on these costs.

Recognize the significance of diversification in investment and risk management.

Describe the effects of changes in market conditions on insurance premiums and market equilibrium.

Understand the concept of adverse selection and how it affects the insurance and labor markets.

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Definitions:

Intellectual Performance

Intellectual performance encompasses cognitive processes such as thinking, reasoning, problem-solving, and memory, reflecting an individual’s capability to learn, make decisions, and adapt to new situations.

Base-10 Number System

The decimal system of numeration based on the use of ten as a base, utilizing digits 0 through 9.

Irregular Number Words

Terms that do not follow the standard pattern or sequence in the linguistic expression of numbers.