The first, second and third terms of an arithmetic progression are $$\(a, \frac{3}{2} a\)$$ and $$\(b\)$$ respectively, where $$\(a\)$$ and $$\(b\)$$ are positive constants. The first, second and third terms of a geometric progression are $$\(a, 18\)$$ and $$\(b+3\)$$ respectively. Find the values of $$\(a\)$$ and $$\(b\)$$. .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... ....................................................................................................................................................
Exam No：9709_s21_qp_12 Year：2021 Question No：8(a)
Answer:
\(
\left(a+b=2 \times \frac{3}{2} a\right) \Rightarrow b=2 a
\)
\(18^{2}=a(b+3)\) OE or 2 correct statements about \(r\) from the GP,
e.g. \(r=\frac{18}{a}\) and \(\mathrm{b}+3=18\) r or \(r^{2}=\frac{b+3}{a}\)
\(
324=a(2 a+3) \Rightarrow 2 a^{2}+3 a-324[=0]
\)
or
\(
b^{2}+3 b-648[=0]
\)
or
\(
6 r^{2}-r-12[=0]
\)
or
\(
4 d^{2}+3 d-162[=0]
\)
\(
(a-12)(2 a+27)[=0]
\)
or
\(
(b-24)(b+27)[=0]
\)
or
\(
(2 r-3)(3 r+4)[=0]
\)
or
\(
(d-6)(4 d+27)[=0]
\)
\(
a=12, b=24
\)
\left(a+b=2 \times \frac{3}{2} a\right) \Rightarrow b=2 a
\)
\(18^{2}=a(b+3)\) OE or 2 correct statements about \(r\) from the GP,
e.g. \(r=\frac{18}{a}\) and \(\mathrm{b}+3=18\) r or \(r^{2}=\frac{b+3}{a}\)
\(
324=a(2 a+3) \Rightarrow 2 a^{2}+3 a-324[=0]
\)
or
\(
b^{2}+3 b-648[=0]
\)
or
\(
6 r^{2}-r-12[=0]
\)
or
\(
4 d^{2}+3 d-162[=0]
\)
\(
(a-12)(2 a+27)[=0]
\)
or
\(
(b-24)(b+27)[=0]
\)
or
\(
(2 r-3)(3 r+4)[=0]
\)
or
\(
(d-6)(4 d+27)[=0]
\)
\(
a=12, b=24
\)
Knowledge points:
1.6.3 use the formulae for the nth term and for the sum of the first n terms to solve problems involving arithmetic or geometric progressions (Including knowledge that numbers a,b,c are 'in arithmetic progression' if 2 b=a+c (or equivalent) and are 'in geometric progression' if (or equivalent) (Questions may involve more than one progression.)
Solution:
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