Proof by induction

Prove by induction that 11 × 7ⁿ - 13ⁿ - 1 is divisible by 3, for all integers n ⩾ 0.

By considering and , make a conjecture for a general formula for in

terms of n and a for any integer n ⩾ 1.

Use induction to prove the formula conjectured in part (a).

The irrational number φ = 1 + plays a significant role in the sequence of Fibonacci

Prove by induction that, for each positive integer n, φⁿ = F_n × φ + F_n-1.

By considering A, A², A³ and A⁴ make a conjecture about the form of the matrix Aⁿ in terms

of n for n ⩾ 1.

Use induction to prove the conjecture made in part (a).

In this question you must show detailed reasoning.

By using an appropriate Maclaurin series prove that if x > 0 then e^x > 1 + x.

Using the inequality in part (b), and by making a suitable choice for t, determine which is

greater, e^π or π^e.

Prove by induction that the sum of the cubes of three consecutive positive integers is divisible by 9.