Question

Prove by the method of induction, for all n ∈ N.

 8 + 17 + 26 + ….. + (9n – 1) = n/2(9n + 7)

Answer 1

Let P(n) ≡ 8 + 17 + 26 +…..+(9n – 1) = n/2(9n + 7), for all n ∈ N.

Step I:

Put n = 1

L.H.S. = 8

R.H.S. = 1/2[9(1) + 7] = 8

∴ L.H.S. = R.H.S.

∴ P(n) is true for n = 1.

Step II:

Let us assume that P(n) is true for n = k.

∴ 8 + 17 + 26 +…..+ (9k – 1) = k/2(9k + 7) …… (i)

Step III:

We have to prove that P(n) is true for n = k + 1,

i.e., 8 + 17 + 26 + …… + [9(k + 1) – 1]

∴ P(n) is true for n = k + 1.

Step IV:

From all the steps above, by the principle of mathematical induction, P(n) is true for all n ∈ N.

∴ 8 + 17 + 26 +…..+ (9n – 1) = n/2 (9n + 7) for all n ∈ N.