A2 June 2025 Paper 1 Q10

AQACurrent spec4 marksInduction

10 Astrid is solving this mathematics problem:

The series \(S_n\) is defined by

\[S_n = 2 + 4 + 6 + \ldots + 2n \quad (n \in \mathbb{Z}, n \geqslant 1)\]

Prove by induction that

\[S_n = n(n + 1)\]

Astrid’s solution is as follows:

Assume the result is true for \(n = k\)

Then

\[\begin{aligned} &S_k = k(k + 1) \\ &S_{k+1} = S_k + 2(k + 1) \\ &S_{k+1} = k(k + 1) + 2(k + 1) \\ &S_{k+1} = (k + 2)(k + 1) \\ &S_{k+1} = (k + 1)((k + 1) + 1) \end{aligned}\]

So the result is also true for \(n = k + 1\)

The result is true for \(n = 1\)

It is true for \(n = k\), and also true for \(n = k + 1\)

Hence, by induction \(S_n = n(n + 1)\) for all integers \(n \geqslant 1\)

(a)
(i) Chloe says that Astrid missed out an essential part of the proof, which could have been written at the start.

Explain what Astrid missed out. [1 mark]

(ii) Write down the working that Astrid missed out. [1 mark]
(b)
(i) One statement in the last three lines of Astrid’s solution is written incorrectly.

Which statement is written incorrectly?

Tick (✓) one box. [1 mark]

  • The result is true for \(n = 1\)
  • It is true for \(n = k\), and also true for \(n = k + 1\)
  • Hence, by induction \(S_n = n(n + 1)\) for all integers \(n \geqslant 1\)
(ii) Write out a correct statement which should replace the incorrect statement identified in part (b)(i) [1 mark]
‹ Previous15 of 56 in InductionNext ›