A2 June 2019 Paper 1 Q9

OCR MEICurrent spec7 marksInduction

9 Prove by induction that \(5^n + 2 \times 11^n\) is divisible by 3 for all positive integers \(n\). [7]

Mark scheme

Scheme	Marks	AO
When \(n = 1\), \(5^1 + 2 \times 11^1 = 27\) div by 3	B1*	2.1
Assume \(u_k = 5^k + 2 \times 11^k\) is div by 3	M1	2.1
\(u_{k+1} = 5^{k+1} + 2 \times 11^{k+1}\)	M1
\(= 5(u_k - 2 \times 11^k) + 22 \times 11^k\)	M1
\(= 5u_k + 12 \times 11^k\)	A1	1.1b
As \(u_k\) div by 3, \(u_{k+1}\) div by 3	A1*	2.2a
So if true for \(n = k\), true for \(n = k + 1\). As true for \(n = 1\), true for all positive integers \(n\)	A1dep	2.4
	[7]

M1: (Assume) or \(5^k + 2 \times 11^k = 3m\)

M1: substituting for \(5^k\)

A1: or \(15m + 12 \times 11^k\)

A1*, A1dep: dep * marks

Scheme	Marks
or \(u_{k+1} = 5^{k+1} + 11(u_k - 5^k)\) \(= 11u_k - 6 \times 5^k\)	M1 A1
or \(u_{k+1} + u_k = 5^{k+1} + 2 \times 11^{k+1} + 5^k + 2 \times 11^k\) \(= 6 \times 5^k + 24 \times 11^k\)	M1 A1

First: substituting for \(11^k\), or \(33m - 6 \times 5^k\); \(5.5^k + 11(3m - 5^k)\)

Second: adding \(u_k\) to \(u_{k+1}\)