Suppose a 1 , a 2 , 2 , a 3 , a 4 be in an arithmetic geometric progression. If the common ratio of the corresponding geometric progression is 2 and the sum of all 5 terms of the arithmetico-geometric progression is 49 2 , then a 4 is equal to ______

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Solution

Q.
Suppose $a_{1}, a_{2}, 2, a_{3}, a_{4}$ be in an arithmetic geometric progression. If the common ratio of the corresponding geometric progression is 2 and the sum of all 5 terms of the arithmetico-geometric progression is $\frac{49}{2}$ , then $a_{4}$ is equal to _______ . [2023]

Ans.
(16)

Let $\frac{a - 2 d}{4}, \frac{a - d}{2}, a, 2 (a + d), 4 (a + 2 d)$ be in A.G.P.

Given that, $a = 2$

$Now, (\frac{1}{2} + 1 + 2 + 4 + 8) + (\frac{- 1}{2} - \frac{1}{2} + 2 + 8) d = \frac{49}{2}$

$\Rightarrow \frac{31}{2} + 9 d = \frac{49}{2} \Rightarrow d = 1$

$Then, a_{4} = 4 (a + 2 d) = 4 (2 + 2) = 16$

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