Question

Question 5 Prove the following by using the principle of mathematical induction for all n∈N: 1.3^2 + 2.3^2 + 3.3^2 + ... + n.3^n = [ (2n-1) 3^(n+1) + 3 ] / 4

Class X1 - Maths -Principle of Mathematical Induction Page 94

Answer 1

1.3 + 2.3² + 3.3³ +.....n.3ⁿ = {(2n-1).3^(n+1) + 3}/4

Let p(n):1.3 + 2.3² + 3.3³ + ....n.3ⁿ = {(2n-1).3^(n+1)+3}/4

step1 :- for n = 1
P(1) : 1.3 = {(2×1-1).3^(1+1)+3}/4
= 1.3
It's true .

step2:- for n = k
p(k): 1.3 + 2.3² +3.3³ +....k.3^k = {(2k-1).3^(k+1) + 3}/4 ------(1)

step3 :- for n= k+1
From eqn (1),
1.3 + 2.3² +3.3³ +....k.3^k = {(2k-1).3^(k+1) + 3}/4
add (k+1).3^(k+1) both sides,


1.3 + 2.3² +3.3³ +....k.3^k + (k+1).3^(k+1) = {(2k-1).3^(k+1) + 3}/4 + (k+1).3^(k+1)
= {(2k-1).3^(k+1) +3 + 4(k+1).3^(k+1)}/4
= {3^(k+1)(2k -1 + 4k + 4 )+3}/4
= {3^(k+1)(6k +3)+3}/4
= [3^{(k+1)+1}.{2(k+1)-1}+3]/4
Hence, p(k+1) is true when p(k) is true . from the mathematical induction, statement is true for all natural numbers.