Sequences - proof by mathematical induction

When a formula has been found for the n-th term of a sequence (by looking for patterns, regression, method of finite differences, ...), there remains the problem of justification or proof. We need to make sure that the formula is correct for every term in the sequence.

An alternative to a direct proof is proof by mathematical induction.

Imagine a ladder that stretches forever.     Are you able to climb onto the first rung?     Are you able to climb from any rung to the one above?     If you answer "yes" to both questions, then you know that you can stand on any rung of the ladder.

Consider the problem of proving the formula for the sum of the first n cube numbers.

This approach is called proof by mathematical induction.

Detailed proof for the sum of cubes problem.

Consider each problem in Sequences - investigations.

Use proof by mathematical induction to justify any rules suggested by the investigations.

Proofs by induction - investigations 1 to 4

Prove by mathematical induction that each of the following is true for all positive integers:

• 7ⁿ + 2 is divisible by 3
• 9ⁿ - 1 is divisible by 8
• n³ + 5n is divisible by 6
• 2ⁿ is greater than or equal to 1 + n

Use mathematical induction to prove Binet’s formula for the n-th term of the Fibonacci sequence.

(Assume the formula is true for n = k and n = k + 1 and show that the formula is true for n = k + 2.)