Provide an Appropriate Response $\sum _ { n = 1 } ^ { \infty } \frac { n ^ { 2 } } { 3 ^ { n - 1 } }$

question 374

Multiple Choice

Provide an appropriate response.
-Find the sum of the series $\sum _ { n = 1 } ^ { \infty } \frac { n ^ { 2 } } { 3 ^ { n - 1 } }$ by expressing $\frac { 1 } { 1 - x }$ as a geometric series, differentiating both sides of the resulting equation with respect to $x$ , multiplying both sides by $x$ , differentiating again, and replacing $x$ by $\frac { 1 } { 3 }$ .

• A) 6
• B) 9
• C) \(\frac { 3 } { 2 }\)
• D) \(\frac { 9 } { 2 }\)

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Definitions:

Older Adults

Individuals who have reached an advanced stage of life, often defined as ages 65 and older, experiencing various physical, social, and cognitive changes.

Falls Prevention

Strategies and actions aimed at reducing the risk and incidence of falls, particularly among elderly populations.

Strength Training

A physical exercise regimen that involves the use of resistance to induce muscular contraction, which builds strength, anaerobic endurance, and muscle size.

Older Adult

An individual in the later stage of life, typically regarded as being 65 years of age or older, focusing on aspects of aging and societal roles.