First slide

Geometric progression

Question

The sum of an infinite number of terms of a G.P. is 20, and the sum of their squares is 100, then the first term of the G.P. is

Moderate

A

5
B

8/5
C

3/5
D

8

Solution

$\begin{array}{l} \frac{a}{1 - r} = 20 and \frac{a^{2}}{1 - r^{2}} = 100 . \\ \Rightarrow \frac{400 (1 - r)^{2}}{1 - r^{2}} = 100 \\ \Rightarrow \frac{1 - r}{1 + r} = \frac{1}{4} \Rightarrow 4 - 4 r = 1 + r \\ \Rightarrow r = 3 / 5 \end{array}$
Thus, $\frac{a}{2 / 5} = 20 \Rightarrow a = 8$

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