Question

Question 1 Prove the following by using the principle of mathematical induction for all n∈N:
1 + 3 + 3^2 + ... 3^(n-1) = (3^n -1) / 2

Class X1 - Maths -Principle of Mathematical Induction Page 94

Answer 1

1 + 3 + 3² + ....3^(n-1) = (3ⁿ-1)/2
Let p(n):1+3 + 3² + .....3^(n-1)=(3ⁿ-1)/2

step1:- for n = 1
P(1): 1 = (3¹-1)/2
It's true .

step 2 :- for n = k
p(k):1 + 3 + 3² +.....3^(k-1)=(3^k-1)/2-----(1)

step3:- for n = k+1
p(k+1):1 + 3 + 3² +.....3^(k-1)+3^k = (3^k-1)/2 [from eqn(1) ]
add both sides, 3^k
1 + 3 + 3² +...3^(k-1)+3^k= (3^k-1)/2 + 3^k
= (3^k-1 + 2×3^k)/2
= (3.3^k-1)/2
= {3^(k+1)-1}/2
Hence, p(k+1)is true when p(k) is true .from the principle of mathematical induction, statement is true for all natural numbers.