2016 H2 Mathematics Paper 1 Question 4

Arithmetic and Geometric Progressions (APs, GPs)

Answers

(i)

r=0.74{r=0.74}

(ii)

3.85(0.74)nb{3.85(0.74)^n b}

Full solutions

(i)

a+3d=br4a+8d=br7a+11d=br14\begin{align} &&\quad a+3d &= br^{4} \\ &&\quad a+8d &= br^{7} \\ &&\quad a+11d &= br^{14} \\ \end{align}
Taking (2)−(1){(2)-(1)},
5d=br7−br4\begin{equation}\qquad 5d = br^{7}-br^{4} \end{equation}
Taking (3)−(1){(3)-(1)},
8d=br14−br4\begin{equation}\qquad 8d = br^{14}-br^{4} \end{equation}
From (4){(4)} and (5){(5)},
br7−br45=br14−br488br7−8br4=5br14−5br45br14−8br7+3br4=05r10−8r3+3=0  ■\begin{gather*} \frac{br^{7}-br^{4}}{5} = \frac{br^{14}-br^{4}}{8} \\ 8br^{7}-8br^{4} = 5br^{14}-5br^{4} \\ 5br^{14} -8br^{7} + 3br^{4} = 0 \\ 5 r^{10} - 8 r^3 + 3 = 0 \; \blacksquare \end{gather*}
Solving with a GC, since ∣r∣<1,{|r|<1,}
r=0.74045=0.74 (2dp)  ■\begin{align*} r &= 0.74045 \\ &= 0.74 \textrm{ (2dp)} \; \blacksquare \end{align*}

(ii)

Sum of terms after, but not including, the n{n}th term
=S∞−Sn=b1−r−b(1−rn)1−r=brn1−r=b(0.74)n1−0.74=3.85(0.74)nb  ■\begin{align*} &= S_{\infty} - S_n \\ &= \frac{b}{1-r} - \frac{b(1-r^n)}{1-r} \\ &= \frac{br^n}{1-r} \\ &= \frac{b(0.74)^n}{1-0.74}\\ &= 3.85(0.74)^n b \; \blacksquare \end{align*}