Prove that 1-2-22-2n-2n-1-1 for all n -

Let #160;pn:1+2+22+...+2n=2n+1 1 #160; #8704;n #8712;NStep #160;I: #160;For #160;n=1,LHS=1+21=3RHS=21+1 1=22 1=4 1=3As, #160;LHS=RHSSo, #160;i ...

You visited us 1 times! Enjoying our articles? Unlock Full Access!

Byju's Answer

Standard XII

Mathematics

Statement

Prove that 1+...

Question

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

Open in App

Solution

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

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

Suggest Corrections

Similar questions

$1 + 2 + 2^{2} + . . . + 2^{n} = 2^{n + 1} - 1$ for all $n ϵ N$ .

Prove that $\frac{1}{2} t a n (\frac{x}{2}) + \frac{1}{4} t a n (\frac{x}{4}) + . . . + \frac{1}{2^{n}} t a n (\frac{x}{2^{n}}) = \frac{1}{2^{n}} c o t + (\frac{x}{2^{n}}) - c o t x$ for all $n ϵ N$ and $0 < x < \frac{x}{2}$ .

Prove that $\frac{(2 n)!}{2^{2 n} (n!)^{2}} \leq \frac{1}{\sqrt{3 n + 1}}$ for all $n ϵ N$ .

Q. Prove that

1 + \frac{2 n}{3} + \frac{2 n (2 n + 2)}{3.6} + \frac{2 n (2 n + 2) (2 n + 4)}{3.6.9} + . . . . .

= 2^{n} {1 + \frac{n}{3} + \frac{n (n + 1)}{3.6} + \frac{n (n + 1) (n + 2)}{3.6.9} + . . . .}

Q. Prove that

\frac{(2 n + 1)!}{n!} = 2^{n} 1.3.5... (2 n - 1) (2 n + 1)

Join BYJU'S Learning Program

Prove that 1 + 2 + 22 + ... + 2n = 2n+1 - 1 for all n ∈ N.

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