Question

Question 4 Prove the following by using the principle of mathematical induction for all n∈N: 1.2.3 + 2.3.4 + … + n(n + 1) (n + 2) = [ n(n+1)(n+2)(n+3) ] / 4

Class X1 - Maths -Principle of Mathematical Induction Page 94

Answer 1

1.2.3 + 2.3.4 + 3.4.5 + .....+n(n+1)(n+2) = n(n+1)(n+2)(n+3)/4

Let p(n):1.2.3 + 2.3.4 +3.4.5 + ....n(n+1)(n+2) = n(n+1)(n+2)(n+3)/4

Step1:- for n = 1
P(1):1.2.3 = 1(1+1)(1+2)(1+3)/4 =1.2.3
It's true.

step2:- for n = k
P(k): 1.2.3 + 2.3.4 + 3.4.5 + .....k(k+1)(k+2) = k(k+1)(k+2)(k+3)/4----------(1)

step3 :- for n = k+1
From eqn (1) ,
1.2.3 + 2.3.4 + 3.4.5 + .....+ k(k+1)(k+2) = k(k+1)(k+2)(k+3)/4
add (k+1)(k+2)(k+3) both sides,

1.2.3 + 2.3.4 + 3.4.5 +.....k(k+1)(k+2)+(k+1)(k+2)(k+3) = k(k+1)(k+2)(k+3)/4 + (k+1)(k+2)(k+3)
= [k(k+1)(k+2)(k+3)+4(k+1)(k+2)(k+3)]/4
= (k+1)(k+2)(k+3)[k+4]/4
=[(k+1){(k+1)+1}{(k+1)+2}{(k+1)+3}]/4
Hence, p(k+1)is true when p(k) is true .from the principle of mathematical induction, statement is true for all natural numbers.