CAIE IGCSE Additional Math 12.3. Recognise arithmetic & geometric progressions Question Bank

Practise deciding whether a sequence is arithmetic or geometric by comparing term-to-term changes and ratios before choosing a method.

[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
B1
$\frac{a+13d}{a+d}=\frac{a+16d}{a+13d}$
oe
M2
$a^2+26ad+169d^2=a^2+17ad+16d^2$
A1
$9ad+153d^2=0$
A1
$9d(a+17d)=0\Rightarrow a=-17d$
oe
A1

(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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12.3. Recognise arithmetic and geometric progressions question 1

Question (a)

Question (b)