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Sequence and Series

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Question : 44 of 86

Marks: +1, -0

The first term of an A.P. is 100 and the sum of whose first 6 terms is 5 times the sum of the next 6 terms, then the c.d. is -

$-10$
10
5
None

Solution:

Concept:

Use sum of n terms formula

S_n = \frac{n}{2}[2a + (n-1)d]

and the given relation between sums.

Explanation:

Given first term

a = 100

, common difference

d

.

Sum of first 6 terms:

S_6 = \frac{6}{2}[2(100) + 5d] = 3(200 + 5d) = 600 + 15d

.

Sum of next 6 terms (terms 7 to 12) =

S_{12} - S_6

.

S_{12} = \frac{12}{2}[2(100) + 11d] = 6(200 + 11d) = 1200 + 66d

.

So sum of next 6 terms =

(1200 + 66d) - (600 + 15d) = 600 + 51d

.

Given

S_6 = 5 \times

(sum of next 6 terms):

600 + 15d = 5(600 + 51d)

.

Simplify:

600 + 15d = 3000 + 255d \implies -240d = 2400 \implies d = -10

.

Answer:

Common difference is

-10

.

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