Enrichment Practice: Mathematical Induction

Outside current Core 9758

Mathematical induction is retained as useful enrichment. These questions must not be treated as required Paper 1 or Paper 2 preparation under the current confirmed syllabus scope.

Answers: Mathematical Induction Answers.

Question 1: Sum Formula

Prove by mathematical induction that

$$1^2+2^2+\cdots +n^2=\frac{n(n+1)(2n+1)}{6}$$

for every positive integer $n$. [8]

Question 2: Divisibility

Prove by mathematical induction that $7^n-2^n$ is divisible by $5$ for every positive integer $n$. [7]

Question 3: Inequality

Prove by mathematical induction that

$$2^n\ge n+1$$

for every integer $n\ge 1$. [6]

Question 4: Recursively Defined Sequence

A sequence is defined by

$$u_1=3,u_{n+1}=2u_n+1.$$

Prove by mathematical induction that

$$u_n=2^{n+1}-1$$

for every integer $n\ge 1$. [7]