JEE Main · 2023 · 10 Apr 2023 · Shift 2 · Sequences and Series

Suppose

a_{1}, a_{2}, 2, a_{3}, a_{4}

be in an arithmetico-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 ______.

Answer: 16

Solution

With first term

A

, common difference

D

and ratio

2

, the five terms are

A, 2 (A + D), 4 (A + 2 D), 8 (A + 3 D), 16 (A + 4 D)

. The middle term equals

2

, so

4 (A + 2 D) = 2

, i.e.

A + 2 D = \frac{1}{2}

. The total

31 A + 98 D = \frac{49}{2}

, together with

A + 2 D = \frac{1}{2}

, gives

D = \frac{1}{4}

and

A = 0

. Therefore

a_{4} = 16 (A + 4 D) = 16 (1) = 16

JEE Main · 2023 · 10 Apr 2023 · Shift 2 · Numerical answer

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