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:

Intimacy Versus Isolation

A stage in Erik Erikson's theory of psychosocial development focusing on the challenge of developing close, committed relationships in early adulthood.

Ego Identity

The sense of self that an individual develops across their life span through social interactions, integrating personal skills, values, and beliefs with societal expectations.

Adulthood

The period in human life when physical growth is completed, and an individual is fully developed, often marked by legal and social responsibilities.

Erikson

A psychoanalyst known for his theory on the psychosocial development of human beings, highlighting eight stages that span from infancy to late adulthood.