A geometric progression has first term $$\(a\)$$, common ratio $$\(r\)$$ and sum to infinity $$\(S\)$$. A second geometric progression has first term $$\(a\)$$, common ratio $$\(R\)$$ and sum to infinity $$\(2 S\)$$. Show that $$\(r=2 R-1\)$$. .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... ....................................................................................................................................................
Exam No：9709_w20_qp_11 Year：2020 Question No：8(a)
Answer:
\(S=\frac{a}{1-r}, \quad 2 S=\frac{a}{1-R}\)
\(\frac{2 a}{1-r}=\frac{a}{1-R}\)
\(2-2 R=1-r \rightarrow r=2 R-1\)
\(\frac{2 a}{1-r}=\frac{a}{1-R}\)
\(2-2 R=1-r \rightarrow r=2 R-1\)
Knowledge points:
1.6.4 use the condition for the convergence of a geometric progression, and the formula for the sum to infinity of a convergent geometric progression.
Solution:
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