June 2022 Paper 2 Q4

Notes

M1: Sum of a GP implied. Allow any \(r\), eg \(r = 0.05\)

A1: Correct equation

A1: Allow 0.487 or 0.488

M1: or \(0.95^{14} = 0.4875\) or 0.487 or 0.488 seen. Can be implied by their answer
ft their equation of form \(a^n = b\) (dep M1 gained and \(b \gt 0\))

A1: cao. Allow \(n = 14\). Allow 14.0. Allow \(\approx 14\)

Alternative method

Scheme	Marks
Sum of GP implied	M1
\(20 + 20 \times r + 20 \times r^2 + \ldots\)	M1
\(20 + 20 \times 0.95 + 20 \times 0.95^2 + \ldots + 20 \times 0.95^{13}\)	A1
\(= 205\) (3 sf)	A1
Number of steps = 14	A1

M1: Attempt add \(\geqslant 10\) terms. Allow any value of \(r\) for this mark

A1: Correct 14 terms added

NB Unsupported correct answer: SC B3

Alternative (incorrect) methods using \(r = 1.05\), or \(\frac{1}{0.95}\) or \(\frac{20}{19}\)

(For info’ only: \(r = \dfrac{1}{0.95}\) or \(\dfrac{20}{19}\) comes from misinterpreting “lowest” to mean “shortest”)

Scheme	Marks
\(20 + 20 \times r + 20 \times r^2 + \ldots\) or \(20 \times \dfrac{1 - r^n}{1 - r}\)	M1
\(20 \times \dfrac{1 - \left(\frac{1}{0.95}\right)^n}{1 - \frac{1}{0.95}} = 205\) or \(20 \times \dfrac{1 - \left(\frac{20}{19}\right)^n}{1 - \frac{20}{19}} = 205\)	A1
\(\left(\dfrac{20}{19}\right)^n = \dfrac{117}{76}\)   or \(1.05^n = 1.51\) or 1.54	A1
\(n = \dfrac{\ln\frac{117}{76}}{\ln\frac{20}{19}}\) or \(\ln_{1.053} 1.539\) or \(\ln_{1.05} 1.51\)	M1
Number of steps = 8 or 9	A0

M1: Allow any value of \(r\) for this mark

A1: oe using 1.05. Correct equation

M1: oe, eg \(\dfrac{\ln 1.539}{\ln 1.053}\) or \(\dfrac{\ln 1.51}{\ln 1.05}\)   ft their “\(\frac{117}{76}\)”
Can be implied by their answer