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JEE Main · 2022 · 26 Jun 2022 · Shift 2 · Sequences and Series

If

a_{1} (> 0), a_{2}, a_{3}, a_{4}, a_{5}

are in a G.P.,

a_{2} + a_{4} = 2 a_{3} + 1

and

3 a_{2} + a_{3} = 2 a_{4}

, then

a_{2} + a_{4} + 2 a_{5}

is equal to ____.

Answer: 40

Solution

Let the G.P. be

a, a r, a r^{2}, a r^{3}, a r^{4}

with

a > 0

.

From

3 a_{2} + a_{3} = 2 a_{4}

:

3 a r + a r^{2} = 2 a r^{3} ⟹ 3 + r = 2 r^{2} ⟹ 2 r^{2} - r - 3 = 0

(2 r - 3) (r + 1) = 0 ⟹ r = \frac{3}{2} or r = - 1

From

a_{2} + a_{4} = 2 a_{3} + 1

:

a r + a r^{3} = 2 a r^{2} + 1 ⟹ a (r + r^{3} - 2 r^{2}) = 1

For

r = \frac{3}{2}

:

a (\frac{3}{2} + \frac{27}{8} - \frac{9}{2}) = a \cdot \frac{3}{8} = 1 ⟹ a = \frac{8}{3}

.

For

r = - 1

:

a (- 1 - 1 + 2) = 0 \neq = 1

, contradiction.

With

a = \frac{8}{3}

and

r = \frac{3}{2}

:

a_{2} + a_{4} + 2 a_{5} = a r + a r^{3} + 2 a r^{4} = 4 + 9 + 27 = 40

JEE Main · 2022 · 26 Jun 2022 · Shift 2 · Integer answer

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