Question

Prove the statement by using principle of mathematical induction:

HM+1(n+2)

1.2+2.3+3.4+ n.(n+1)=
........n.(n+1)=

n.(n+1) (n+2)
________
3​

Answer 1

Answer:

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Answer 2

let p(n):=1*2+2*3+3*4.....+n(n+1)=n(n+1)(n+2)/3

LHS

p(1)=1(1+1)=1(2) = 2

RHS

p(1)=1(1+1)(1+2)/3 = 1(2)(3)/3 = 2

LHS =RHS

therefore p(1) is true .

Assume that the statement is true for n=k,

p(k)=1*2+2*3+3*4+.....+k(k+1)=k(k+1)(k+2)/3

We need to show that the statement is true for k+1.

p(k+1)=1*2+2*3+.....+k(k+1)+k+1(k+1+1)

p(k+1)= p(k)+ (k+1)(k+2)

p(k+1)= k(k+1)(k+2)/3 +(k+1)(k+2)

taking (k+1)(k+2) as common

We can conclude that p(k+1) is true.

By the principle of mathematical induction, it is proved that p(n):=1*2+2*3+..+n(n+1)=n(n+1)(n+2)/3 is true.