Answer: (C) 4n-3n+1
For n = 1,
8n + 1 = 81 + 1 = 9 divisible by 9
10n + 1 = 101 + 1 = 11 not divisible by 9
4n – 3n – 1 = 4 – 3 – 1 = 0 divisible by 9
32n + 3n + 1 = 13 not divisible by 9
For n = 2
8n + 1 = 82 + 1 = 65 not divisible by 9
4n – 3n – 1 = 42 – 3 × 2 – 1 = 16 – 6 – 1 = 9 divisible by 9
∴ We need to prove 4n – 3n – 1 to be divisible by 9 ∀ n∈N.
using mathematical induction.
Let T(n): 4n – 3n – 1 is divisible by 9,
Basic Step:
T(1) = 0 which is divisible by 9
⇒ T(1) is true.
Induction Step:
Assume T(k) to be true, i.e.,
4k – 3k – 1 is divisible by 9 k∈N
⇒ 4k – 3k – 1 = 9m, m∈N ...(i)
∴ 4k + 1 – 3(k + 1) – 1 = 4.4k – 3k – 3 – 1 = 4.4k – 3k – 4
= 4(4k – 3k – 1) + 9k = 4.9m + 9k = 9 (4m + k)
⇒ 4k + 1 – 3 (k + 1) – 1 is divisible by 9
⇒ T (k + 1) is true whenever T(k) is true, k∈N
⇒ 4n + 3n – 1 is divisible by 9 ∀ n∈N