Department of Pre-University Education, KarnatakaPUC Karnataka Science Class 11

# Prove that 1 + 2 + 22 + ... + 2n = 2n+1 - 1 for All N ∈ N . - Mathematics

Prove that 1 + 2 + 22 + ... + 2n = 2n+1 - 1 for all $\in$ N .

#### Solution

$Let p\left( n \right): 1 + 2 + 2^2 + . . . + 2^n = 2^{n + 1} - 1 \forall n \in N$
$\text{ Step I: For } n = 1,$
$LHS = 1 + 2^1 = 3$
$RHS = 2^{1 + 1} - 1 = 2^2 - 1 = 4 - 1 = 3$
$As, LHS = RHS$
$\text{ So, it is true for n } = 1 .$
$\text{ Step II: For n } = k,$
$\text{ Let } p\left( k \right): 1 + 2 + 2^2 + . . . + 2^k = 2^{k + 1} - 1\text{ be true } \forall k \in N$
$\text{ Step III: For } n = k + 1,$
$LHS = 1 + 2 + 2^2 + . . . + 2^k + 2^{k + 1}$
$= 2^{k + 1} - 1 + 2^{k + 1} \left(\text{ Using step } II \right)$
$= 2 \times 2^{k + 1} - 1$
$= 2^{k + 1 + 1} - 1$
$= 2^{k + 2} - 1$
$RHS = 2^\left( k + 1 \right) + 1 - 1 = 2^{k + 2} - 1$
$As, LHS = RHS$
$\text{ So, it is also true for n } = k + 1 .$

Hence, 1 + 2 + 22 + ... + 2n = 2n+1 - 1 for all n $\in$ N .

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#### APPEARS IN

RD Sharma Class 11 Mathematics Textbook
Chapter 12 Mathematical Induction
Exercise 12.2 | Q 30 | Page 28