S? be a series which starts with a (positive odd integer) and has a negative common difference. The sum of first 4

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Solution

Q.
$S ₙ$ be a series which starts with a (positive odd integer) and has a negative common difference. The sum of first 4 terms is 16 and sum of first 6 terms is 12. Which of the following statements are true

(i) First term of the series is 7.
(ii) Common difference is –2.
(iii) Sum will be positive for first 10 terms.
(iv) Sum will be positive for first 8 terms.

Choose the correct option from the following:

1 (i) and (iii)

2 (i) and (ii)

3 (ii) and (iii)

4 (iii) and (iv)

Ans.
(2)

Given a is positive odd integer and d is negative

$S ₄ = 16 a n d S ₆ = 12 W e k n o w t h a t S ₙ = n / 2 (2 a + (n - 1) d) S ₄ = 4 / 2 (2 a + (4 - 1) d) \Rightarrow 2 (2 a + 3 d) = 16 \Rightarrow 2 a + 3 d = 8 . . . (i) A l s o, S ₆ = 6 / 2 (2 a + (6 - 1) d) = 12 \Rightarrow 3 (2 a + 5 d) = 12 \Rightarrow 2 a + 5 d = 4 . . . (i i) S o l v i n g (i) a n d (i i) 2 a + 3 d = 8 2 a + 5 d = 4 - - - - - 2 d = 4 \Rightarrow d = - 2 P u t v a l u e o f d i n e q u a t i o n (i) 2 a + 3 (- 2) = 8 \Rightarrow 2 a - 6 = 8 \Rightarrow 2 a = 14 \Rightarrow a = 7 S ₁ ₀ = 10 / 2 (2 (7) + (9) (- 2)) = 5 (14 - 18) = - 20 (n e g a t i v e s u m) S ₈ = 8 / 2 (2 (7) + (7) (- 2)) = 4 (14 - 14) = 0 H e n c e, (i) a n d (i i) a r e c o r r e c t$

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