Question

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

(2⁴ⁿ – 1) is divisible by 15

Answer 1

(2⁴ⁿ – 1) is divisible by 15 if and only if (2⁴ⁿ – 1) is a multiple of 15.

Let P(n) ≡ (2⁴ⁿ – 1) = 15m, where m ∈ N.

Step I:

Put n = 1

∴ 2⁴⁽¹⁾  – 1 = 16 – 1 = 15

∴ (2⁴ⁿ – 1) is a multiple of 15.

∴ P(n) is true for n = 1.

Step II:

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

∴ 2^4k  – 1 = 15a, where a ∈ N

∴ 2^4k  = 15a + 1 …..(i)

Step III:

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

i.e., to prove that

∴ 2^4(k+1)  – 1 = 15b, where b ∈ N

∴ P(k + 1) = 2^4(k+1) – 1 = 2^4k+4  – 1

= 2^4k . 2⁴ – 1

= 16 . (2^4k) – 1

= 16(15a + 1) – 1 …..[From (i)]

= 240a + 16 – 1

= 240a + 15

= 15(16a + 1)

= 15b, where b = (16a + 1) ∈ N

∴ 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.

∴ (2⁴ⁿ – 1) is divisible by 15, for all n ∈ N.