A sum, an induction proof and an application

Choose your method, connect the parts and explain your conclusions. These original questions include written reasoning and sketches as well as exact answer checks.

Suggested marks guide how much working to show. The site does not automatically award examination marks for proofs, diagrams or methods. Use scaffolded fluency practice when you need a hint first.

A-level core. Course filters use selected objective mappings; none of these tasks establishes complete paper coverage.

Before you start

Proof by induction. Read the relevant method, then decide which parts you can solve without hints.

Check a misconception first · Try a connected problem

Read an original example question

Let $S_n=1+3+\cdots+(2n-1)$ .

Part a

Prove that $S_{n}$ = $n^{2}$ for every integer n ≥ 1.

Part b

Hence find the number of terms needed for a sum of 100.

Part c

A proposed induction step adds 2k−1. Explain the mistake.

Model solution and review criteria

Part a

At n=1, $S_{1}$ =1= $1^{2}$ . Assume $S_{k}$ = $k^{2}$ . Then $S_{k+1}$ = $S_{k}$ +(2k+1)= $k^{2}$ +2k+1=(k+1)². The base case and arbitrary-step implication prove the claim.

Check the base case.
State the arbitrary k hypothesis.
Use the next summand and complete the square.
Conclude over the stated domain.

Part b

$n^{2}$ = 100 and n is a positive integer, so n = 10.

Use the proven formula and discard a negative term count.

Part c

The hypothesis already includes 2k−1. The next term is 2(k+1)−1=2k+1. Adding the previous term does not produce $S_{k+1}$ .

Identify exactly which summand comes next.

Compare your own reasoning. No automatic examination marks are awarded.