Use the Principle of Mathematical Induction to Prove That $1 - 2 + 2 ^ { 2 } - 2 ^ { 3 } + \cdots + ( - 1 ) ^ { n } 2 ^ { n } = \frac { 2 ^ { n + 1 } ( - 1 ) ^ { n } + 1 } { 3 }$

question 17

Short Answer

Use the Principle of Mathematical Induction to prove that $1 - 2 + 2 ^ { 2 } - 2 ^ { 3 } + \cdots + ( - 1 ) ^ { n } 2 ^ { n } = \frac { 2 ^ { n + 1 } ( - 1 ) ^ { n } + 1 } { 3 }$ for all positive integers n .

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Definitions:

Substitute

A good or service that can be used in place of another, allowing consumers to switch between them based on price, preference, or availability.

Equilibrium

A situation where the supply and demand in the market are equal, leading to stable prices.

Margarine

A spread used as a substitute for butter, made from vegetable oils or animal fats.

Producer Surplus

The difference between what producers are willing to sell a good for and the price they actually receive.