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 .

Name: 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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Question

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 .

Quizplus · Accepted Answer

The Answer is

P ( 1 ) : 1 - 2 = \frac { 2 ^ { 2 } ( - 1 ) + 1 } { 3 }

, which is true since both sides are equal to

- 1 . \quad P ( k ) ightarrow P ( k + 1 ) : 1 - 2 + 2 ^ { 2 } +

\begin{array} { l } \cdots + ( - 1 ) ^ { k + 1 } 2 ^ { k + 1 } = \frac { 2 ^ { k + 1 } ( - 1 ) ^ { k } + 1 } { 3 } + ( - 1 ) ^ { k + 1 } 2 ^ { k + 1 } = \frac { 2 ^ { k + 1 } ( - 1 ) ^ { k } + 1 + 3 ( - 1 ) ^ { k + 1 } 2 ^ { k + 1 } } { 3 } \= \frac { 2 ^ { k + 1 } ( - 1 ) ^ { k } ( 1 + 3 ( - 1 ) ) + 1 } { 3 } = \frac { 2 ^ { k + 1 } ( - 1 ) ^ { k } ( - 2 ) + 1 } { 3 } = \frac { 2 ^ { k + 2 } ( - 1 ) ^ { k + 1 } + 1 } { 3 } .\end{array}

Use the Principle of Mathematical Induction to Prove That 1−2+22−23+⋯+(−1)n2n=2n+1(−1)n+131 - 2 + 2 ^ { 2 } - 2 ^ { 3 } + \cdots + ( - 1 ) ^ { n } 2 ^ { n } = \frac { 2 ^ { n + 1 } ( - 1 ) ^ { n } + 1 } { 3 }1−2+22−23+⋯+(−1)n2n=32n+1(−1)n+1​

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