AS June 2018 Q4

EdexcelCurrent spec10 marksRecurrence Relations

4. A village has an expected population growth rate (birth rate minus death rate) of \(r\)% per year.
In addition, \(N\) people are expected to move into the village each year.
The expected population of the village is modelled by

\[u_{n+1} = 1.02\,u_n + 50,\]

where \(u_n\) is the expected population of the village \(n\) years from now.

(a) State

(i) the value of \(r\),

(ii) the value of \(N\). (2)

Given that the population 1 year from now is expected to be 560

(b) solve the recurrence relation for \(u_n\) (5)

(c) Hence determine, using algebra, the number of years from now when the model predicts that the population of the village will first be greater than 3000 (3)

Scheme	Marks	AO
\(r = 2\) \(N = 50\)	B1 B1	3.4 1.1b
	(2)

Mark scheme (b)

Scheme	Marks	AO
(aux equation \(m - 1.02 = 0 \Rightarrow\)) complementary function is \(A(1.02)^n\)	B1	1.1b
Consider a trial solution of the form \(u_n = \lambda\) so \(\lambda - 1.02\lambda = 50\) \(\Rightarrow \lambda = \ldots\)	M1	1.1b
General solution is \(u_n = A(1.02)^n - 2500\)	A1	1.1b
\(n = 1,\ u_1 = 560 \Rightarrow A = \ldots\)	M1	3.4
\(u_n = 3000(1.02)^n - 2500\)	A1	1.1b
	(5)

Notes

B1: cao

M1: substituting their trial solution into the recurrence relation in an attempt to find their \(\lambda\)

A1: cao for the general solution

M1: using the conditions in the model to calculate \(A\)

A1: cao for the particular solution

Alternative approach for (b)

B1: \((1.02)^n u_0\)

M1: attempt sum of GP with \(a = 50\) and \(r = 1.02\) \(\left(u_n = \ldots + \dfrac{50\left(1 - 1.02^n\right)}{1 - 1.02}\right)\)

A1: general solution is \(u_n = (1.02)^n(u_0 + 2500) - 2500\) (or equivalent)

M1: Uses \(u_1 = 560\) to find \(u_0\) (e.g. \(560 = 1.02u_0 + 50 \Rightarrow u_0 = \ldots\))

A1: \(u_n = 3000(1.02)^n - 2500\)

Mark scheme (c)

Scheme	Marks	AO
\(3000(1.02)^n - 2500 \gt 3000\)	M1	1.1b
\((1.02)^n \gt \dfrac{11}{6} \Rightarrow n\log(1.02) \gt \log\left(\dfrac{11}{6}\right)\)	M1	1.1b
\(n \gt 30.6088\ldots \Rightarrow n = 31\)	A1	1.1b
	(3)
	(10 marks)

Notes

M1: sets their particular solution greater than 3000 (condone equals) – their particular solution must be of the correct form \(\left(u_n = c(1.02)^n \pm d\right)\)

M1: dependent on previous M mark – re-arranging and correctly applies the process of taking logs for their particular solution

A1: cao (allow correct answer to 3 significant figures or 31)