CAIE IGCSE Additional Math 12.3 Recognise Arithmetic and Geometric Progressions Questions

Practise distinguishing arithmetic from geometric progressions by testing common differences and ratios in Cambridge International Additional Mathematics sequences.

Exam points

Recognise whether a sequence is arithmetic or geometric by comparing successive terms and identifying the common difference or ratio.

[Maximum number: 8]

An arithmetic progression, A, has first term a and common difference d. The 2nd, 14th and 17th terms of A form the first three terms of a convergent geometric progression, G, with common ratio r .

(a)

Given that $d\neq0$ , find two expressions for r in terms of a and d and hence show that a=-17d.

[ 6 ]

$a+d,\quad a+13d,\quad a+16d$
soi
$\frac{a+13d}{a+d}=\frac{a+16d}{a+13d}$
oe
$a^2+26ad+169d^2=a^2+17ad+16d^2$ $9ad+153d^2=0$ $9d(a+17d)=0\Rightarrow a=-17d$
oe

(b)

Find the value of r .

[ 2 ]

r=0.25
oe
M1 for $\frac{-17d+13d}{-17d+d}$ or $\frac{-17d+16d}{-17d+13d}$

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CAIE IGCSE Additional Math 12.3 Recognise Arithmetic and Geometric Progressions Questions question 1

Question (a)

Question (b)