Prove by induction: $2^1+2^2+2^3+........+2^n=2(2^n−1)$

Question Collected from Telegram Group.

Solution:

P(n):2¹+2²+2³+...+2ⁿ=2(2ⁿ−1)

Step 1: Prove that the statement is true for n=1

P(1):2¹=2(2¹−1)

P(1):2=2

Hence, the statement is true for n=1

Step 2: Assume that the statement is true for n=k

Let us assume that the below statement is true:

P(k):2+2²+...+2^k=2(2^k−1)

Step 3: Prove that the statement is true for n=k+1

We need to prove that:

2+2²...+2^k+1=2(2^k+1−1)

LHS=2+2²+...+2^k+2^k+1

=2(2^k−1)+2^k+1

=2(2^k−1+2^k)

=2(2.2^k−1)

=2(2^k+1−1)

=RHS

Therefore, P(n) is true for all values of n by principle of mathematical induction.