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June 2021 p12 q8
817
The first, second and third terms of an arithmetic progression are \(a\), \(\frac{3}{2}a\) and \(b\) respectively, where \(a\) and \(b\) are positive constants. The first, second and third terms of a geometric progression are \(a\), 18 and \(b + 3\) respectively.
(a) Find the values of \(a\) and \(b\).
(b) Find the sum of the first 20 terms of the arithmetic progression.
Solution
(a) For the arithmetic progression, the second term is \(\frac{3}{2}a\) and the third term is \(b\). The common difference \(d\) is given by:
\(d = \frac{3}{2}a - a = \frac{a}{2}\)
\(b = \frac{3}{2}a + \frac{a}{2} = 2a\)
For the geometric progression, the second term is 18 and the third term is \(b + 3\). The common ratio \(r\) is given by:
\(r = \frac{18}{a}\) and \(b + 3 = 18r\)
Substituting \(b = 2a\) into \(b + 3 = 18r\), we get:
\(2a + 3 = 18 \times \frac{18}{a}\)
\(2a^2 + 3a - 324 = 0\)
Solving this quadratic equation, we find \(a = 12\) and \(b = 24\).
(b) The common difference \(d = 6\) (since \(d = \frac{a}{2} = 6\)). The sum of the first 20 terms \(S_{20}\) is given by:
