2015 H2 Mathematics Paper 1 Question 8

Arithmetic and Geometric Progressions (APs, GPs)

Answers

(i)

[59,77]{\left[ 59, 77 \right]}

(ii)

[63.9,74.4]{\left[ 63.9, 74.4 \right]}

(iii)

11 s{11 \textrm{ s}}

Full solutions

(i)

5400≤Sn≤63005400 \leq S_n \leq 6300
5400≤n2(2a+(n−1)d)≤63005400≤502(2T+(50−1)(2))≤6300\begin{alignat*}{2} 5400 &\leq \frac{n}{2} \Big( 2a + \left( n - 1 \right) d \Big) &\leq 6300 \\ 5400 &\leq \frac{50}{2} \Big( 2T + (50-1)(2) \Big) &\leq 6300 \\ \end{alignat*}
5400≤50T+2450≤630059≤T≤77\begin{gather*} 5400 \leq 50T + 2450 \leq 6300 \\ 59 \leq T \leq 77 \\ \end{gather*}
Set of values of T=[59,77]  ■{T = \left[ 59, 77 \right] \; \blacksquare}

(ii)

5400≤Sn≤63005400 \leq S_n \leq 6300
5400≤a(rn−1)r−1≤63005400≤t(1.0250−1)1.02−1≤6300\begin{alignat*}{2} 5400 &\leq \frac{a\left(r^{n}-1\right)}{r-1} &\leq 6300 \\ 5400 &\leq \frac{t\Big( 1.02^{50} - 1 \Big)}{1.02-1} &\leq 6300 \\ \end{alignat*}
63.845≤t≤74.48663.845 \leq t \leq 74.486
Set of values of t=[63.9,74.4]  ■{t = \left[ 63.9, 74.4 \right] \; \blacksquare}

(iii)

Let T=59{T=59} and t=63.845{t=63.845}
For athlete A,{A,}
u50=a+(n−1)d=59+(50−1)2=157\begin{align*} u_{50} &= a + \left( n - 1 \right) d \\ &= 59 + \left( 50 - 1 \right) 2 \\ &= 157 \end{align*}
For athlete B,{B,}
u50=arn−1=74.486(1.02)50−1=168.47\begin{align*} u_{50} &= ar^{n-1} \\ &= 74.486\left(1.02\right)^{50-1} \\ &= 168.47 \end{align*}
Difference in athletes' times
=168.47−157=11 s (nearest second)  ■\begin{align*} &= 168.47 - 157 \\ &= 11 \textrm{ s (nearest second)} \; \blacksquare \end{align*}