First slide

Geometric progression

Question

In a sequence of 21 terms the first 11 terms are in A.P. with common difference 2 and the last 11 terms are in G.P. with common ratio 2 . If the middle term of the A.P. is equal to the middle term of the G.P., then the middle term of the entire sequence is

Moderate

A

$\frac{- 10}{31}$
B

$\frac{- 630}{31}$
C

$\frac{- 32}{31}$
D

$\frac{- 320}{31}$

Solution

$Here common difference d = 2, common ratio = 2$
$Given, middle term of the A . P = middle term of the G.P$
$\Rightarrow a + (6 - 1) 2 = [a + (11 - 1) 2] 2^{5}$
$\Rightarrow a = - \frac{630}{31}$
$∴ m i d d l e t e r m = a + 10 d = a + 20$
$= - \frac{630}{31} + 20$
$= - \frac{10}{31}$

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