A geometric progression has first term $$\(a\)$$, common ratio $$\(r\)$$ and sum to infinity $$\(S\)$$. A second geometric progression has first term $$\(a\)$$, common ratio $$\(R\)$$ and sum to infinity $$\(2 S\)$$. It is now given that the 3rd term of the first progression is equal to the 2 nd term of the second progression. Express $$\(S\)$$ in terms of $$\(a\)$$. .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... ....................................................................................................................................................
Exam No：9709_w20_qp_11 Year：2020 Question No：8(b)
Answer:
$\begin{array}{l}a r^{2}=a R \rightarrow(a)(2 R-1)^{2}=R(a) \\ 4 R^{2}-5 R+1(=0) \rightarrow(4 R-1)(R-1)(=0) \\ R=\frac{1}{4} \\ S=\frac{2 a}{3}\end{array}$
Alternative method for question $\mathbf{8 ( b )}$
$\begin{array}{l}a r^{2}=a R \rightarrow(a) r^{2}=1 / 2(r+1)(a) \\ 2 r^{2}-r-1(=0) \rightarrow(2 r+1)(r-1)(=0) \\ r=-\frac{1}{2} \\ S=\frac{2 a}{3}\end{array}$
Alternative method for question $\mathbf{8 ( b )}$
$\begin{array}{l}a r^{2}=a R \rightarrow(a) r^{2}=1 / 2(r+1)(a) \\ 2 r^{2}-r-1(=0) \rightarrow(2 r+1)(r-1)(=0) \\ r=-\frac{1}{2} \\ S=\frac{2 a}{3}\end{array}$
Knowledge points:
1.1.3 solve quadratic equations, and quadratic inequalities, in one unknown (By factorising, completing the square and using the formula.)
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:
Download APP for more features
1. Tons of answers.
2. Smarter Al tools enhance your learning journey.
IOS
Download
Download
Android
Download
Download
Google Play
Download
Download