Question

Prove by method of induction, for all n ∈ N
$\rm 1.2+2.3+3.4+... +n(n+1)= \frac{n}{3}(n+1)(n+2)$

Answer 1

Answer:

1.2  + 2.3  + 3.4  +........................+n(n+1) =  (n/3)(n+1)(n+2)

Step-by-step explanation:

1.2  + 2.3  + 3.4  +........................+n(n+1) =  (n/3)(n+1)(n+2)

p(1) = 1.2  =  (1/3)(1+1)(1+2)

=> 2 = 2

p(2) = 1.2 + 2.3 = (2/3)(2+1)(2+2)

=> 2 + 6 = (2/3)(3)(4)

=> 8 = 8

let assume p(k) is true

then

1.2  + 2.3  + 3.4  +........................+k(k+1) =  (k/3)(k+1)(k+2)

to be proved

p(k + 1) = 1.2  + 2.3  + 3.4  +....................+k(k+1)  + (k+1)(k+2) = ((k+1)/3)(k+2)(k+3)

LHS =

1.2  + 2.3  + 3.4  +....................+k(k+1)  + (k+1)(k+2)

= (k/3)(k+1)(k+2) +  (k+1)(k+2)

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

=  (k+1)(k+2) (k + 3)/3

=  ((k+1)/3)(k+2)(k+3)

= RHS

Hence

1.2  + 2.3  + 3.4  +........................+n(n+1) =  (n/3)(n+1)(n+2)