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Proof by mathematical induction

Use the princ...

Question

Use the principle of mathematical induction to show that :
$2 + 2^{2} + . . . . . + 2^{n} = 2^{n + 1} - 2$ for every natural number $n .$

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Solution

$\begin{matrix} p (n) : 2 + 2^{2} + . . . . . . . . . . . . . . + 2^{n} = 2^{n + 1} - 2 W h e n, n = 1 L . H . S = 2 a n d R . H . S = 2 s o, f o r n = 1 p (n) i s t r u e . L e t P (R) : 2 + 2^{2} + . . . {.2}^{k} = 2^{k + 1} - 2 T h e r e f o r e, p (k + 1) : 2 + 2^{2} + . . . {.2}^{k + 1} = 2^{k + 2} - 2 = 2^{k + 1} .2 - 2 = 2 (2^{k + 1} - 1) S o, p (k + 1) i s a l s o t r u e . H e n c e b y t h e p r i n c i p l e o f m a t h e m a t i c a l i n d u c t i o n, p (n) i s t r u e f o r a l l n \in N . \end{matrix}$

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