In an increasing geometric progression of positive terms, the sum of the second and sixth terms is 70 3 and the product of the third and fifth terms is 49. Then the sum of the 4th, 6th, and 8th terms is equal to [2024]

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Solution

Q.
In an increasing geometric progression of positive terms, the sum of the second and sixth terms is $\frac{70}{3}$ and the product of the third and fifth terms is 49. Then the sum of the 4th, 6th, and 8th terms is equal to [2024]

1 78

2 96

3 84

4 91

Ans.
(4)

Let the G.P. be $\frac{a}{r^{3}}, \frac{a}{r^{2}}, \frac{a}{r}, a, a r, a r^{2}, a r^{3}, a r^{4}$

Now, $\frac{a}{r^{2}} + a r^{2} = \frac{70}{3}$ ...(i)

and $\frac{a}{r} \times a r = 49 \Rightarrow a^{2} = 49 \Rightarrow a = 7$ (G.P. is increasing)

Now, $\frac{7}{r^{2}} + 7 r^{2} = \frac{70}{3}$ [Using (i)]

$\Rightarrow 3 r^{4} - 10 r^{2} + 3 = 0$

$\Rightarrow (3 r^{2} - 1) (r^{2} - 3) = 0 \Rightarrow r^{2} = \frac{1}{3} or r^{2} = 3$

As the G.P. is increasing, $∴$ $r^{2} = 3 \Rightarrow r = \sqrt{3}$

Now, $a + a r^{2} + a r^{4} = 7 + 7 (3) + 7 (9) = 91$

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