Question

Prove the following by PMI for all n ∈ N.
1. 1.3 + 2.4 + 3.5 + ⋯ ... ... =n(n+1)(2n+7)/6

Answer 1

Step-by-step explanation:

Suppose P (n): 1.3 + 2.4 + 3.5 + … + n. (n + 2) = 1/6 n(n + 1) (2n + 7)

Now let us check for n = 1,

P (1): 1.3 = 1/6 × 1 × 2 × 9

: 3 = 3

P (n) is true for n = 1.

Then, let us check for P (n) is true for n = k, and have to prove that P (k + 1) is true.

P (k): 1.3 + 2.4 + 3.5 + … + k. (k + 2) = 1/6 k(k + 1) (2k + 7) … (i)

Therefore,

1.3 + 2.4 + 3.5 + … + k. (k + 2) + (k + 1) (k + 3)

Then, substituting the value of P (k) we get,

= 1/6 k (k + 1) (2k + 7) + (k + 1) (k + 3) by using equation (i)

= (k + 1) [{k(2k + 7)/6} + {(k + 3)/1}]

= (k + 1) [(2k2 + 7k + 6k + 18)]/6

= (k + 1) [2k2 + 13k + 18]/6

= (k + 1) [2k2 + 9k + 4k + 18]/6

= (k + 1) [2k(k + 2) + 9(k + 2)]/6

= (k + 1) [(2k + 9) (k + 2)]/6

= 1/6 (k + 1) (k + 2) (2k + 9)

P (n) is true for n = k + 1

Thus, P (n) is true for all n ∈ N