Question

By Principle of Mathematical Induction, prove that for all n ∈ N. Show that 3.5^{2n + 1} + 2^{3n + 1} is divisible by 17 for all n ∈ N.

Answer 1

Let S(n) be the statement that 3.5^{2n + 1} + 2^{3n + 1} is divisible by 17

If n = 1, then given expression = 3 * 5³ + 2⁴ + 375 + 16 = 391 = 17 * 23, divisible by 17.

S(1) is true

Assume that S(k) is true.

3.5^{2k + 1} + 2^{3k + 1} is divisible by 17.

 3.5^{2k = 1} + 2^{3k + 1} = 17m where m  N

3.5^{2(k + 1) + 1} + 2^{3(k + 1) + 1} = 3.5^{2k + 1} * 5² + 2^{3k + 1} * 2³ = 25(17m – 2^3k+1) + 8.2^{3k + 1}

= 425m – 25.2^{3k + 1} + 8.2^{3k + 1} = 425m – 17.2^{3k + 1} = 17(25m – 2^{3k + 1}), divisible by 17

S(k + 1) is true

By Principle of Mathematical Induction S(n) is true for all n ∈ N.

3.5^{2n + 1} + 2^{3n + 1} is divisible by 17 for all n ∈ N