First slide

Geometric progression

Question

The sum of first four terms of a geometric progression (G.P.) is $\frac{65}{12}$ and the sum of their respective reciprocals is $\frac{65}{18}$ . If the product of first three terms of the G.P. is 1, and the third term is $α$ , then $2 α$ is___

Moderate

Solution

$\begin{array}{l} a + ar + {ar}^{2} + {ar}^{3} = \frac{65}{12}, \frac{1}{a} + \frac{1}{ar} + \frac{1}{{ar}^{2}} + \frac{1}{{ar}^{3}} = \frac{65}{18} \cdot a^{3} r^{3} = 1 \Rightarrow ar = 1 \\ a^{3} r^{3} = 1 \Rightarrow ar = 1 \\ \frac{1 + r + r^{2} + r^{3}}{r} = \frac{65}{12}, \frac{r^{3} + r^{2} + r + 1}{r^{2}} = \frac{65}{18} \\ r = \frac{18}{12} = \frac{3}{2} \Rightarrow α = {ar}^{2} = 1 \cdot \frac{3}{2} = \frac{3}{2} \Rightarrow 2 α = 3 \end{array}$

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