Prove That $\frac { 1 } { 2 } + \frac { 2 } { 4 } + \frac { 3 } { 8 } + \cdots + \frac { n } { 2 ^ { n } } = \frac { 2 ^ { n + 1 } - 2 - n } { 2 ^ { n } }$

question 38

Short Answer

Prove that $\frac { 1 } { 2 } + \frac { 2 } { 4 } + \frac { 3 } { 8 } + \cdots + \frac { n } { 2 ^ { n } } = \frac { 2 ^ { n + 1 } - 2 - n } { 2 ^ { n } }$ for all $n \geq 1$

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Definitions:

Gross Domestic Product (GDP)

The aggregate value of all products and services generated inside a nation's boundaries over a particular time frame, serving as a comprehensive indicator of economic performance.

Gross Domestic Product (GDP)

The sum total of the market or economic value of all products and services finalized within a country's geographical limits during a defined period.

Gross Domestic Product (GDP)

The full valuation, in market or monetary terms, of all end goods and services generated within the confines of a country's borders in a particular time span.

Household Production

The process by which families produce goods and services for their own use, typically including tasks such as cooking, cleaning, and childcare.