Question

Question 6 Prove the following by using the principle of mathematical induction for all n∈N: 1.2 + 2.3 + 3.4 + ... + n.(n+1) = [ n(n+1)(n+2) ] / 3

Class X1 - Maths -Principle of Mathematical Induction Page 94

Answer 1

1.2 + 2.3 + 3.4 + ......+ n(n+1) = [n(n+1)(n+2)/3]
Let p(n): 1.2 + 2.3 +3.4 + ......n(n+1) = [n(n+1)(n+2)/3]

step1:- for n = 1
p(1): 1.2 = [1×(1+1)(1+2)/3]
= 1.2
It's true.

step2:- for n = k
p(k): 1.2 + 2.3 + 3.4 + ......k(k+1) = [k(k+1)(k+2)/3] ----------(1)

step3:- for n = k+1
From eqn (1)
1.2 + 2.3 + 3.4 +....k(k+1) = [k(k+1)(k+2)/3]
add (k+1)(k+2) both sides,
1.2 + 2.3 + 3.4 + .. .+ k(k+1) +(k+1)(k+2) = [k(k+1)(k+2)/3]+(k+1)(k+2)
= {k(k+1)(k+2)+3(k+1)(k+2)}/3
= (k+1)(k+2){k+3}/3
= [(k+1){(k+1)+1}{(k+1)+2}/3]
Hence, p(k+1) is true when p(k) is true . from the mathematical induction, statement is true for all natural numbers.