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The sum of first 16 terms of an A.P is 112 and sum of its next

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The sum of first $16$ terms of an $A . P$ is $112$ and sum of its next fourteen terms is $518$ . Find the $A . P$ .

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Elijah Discussed

The sum of first

16

terms of an

A . P

112

and sum of its next fourteen terms is

518

. Find the

A . P

6 mins ago

Discuss this question LIVE

6 mins ago

Text solutionVerified

Given

S_{16} = 112

sum of next 14 terms = 518

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

S_{16} = \frac{16}{2} [2 a + (16 - 1) d]

112 = 8 [2 a + 15 d]

14 = 2 a + 15 d \dots (1)

sum of next 14 terms is 518

sum of (16 + 14) terms

(S_{30}) = 112 + 518

S_{30} = 630

S_{30} = \frac{30}{2} [2 a + (30 - 1) d]

630 = 15 [2 a + 29 d]

42 = 2 a + 29 d \dots (2)

Subtracting equation (1) from (2)

42 = 2 a + 29 d

14 = 2 a + 15 d

- - -

_____________

28 = 14 d .

d = 28/14

d = 2.

Substitute value of d in equation (1)

14 = 2 a + 15 d

14 = 2 a + 15 (2)

14 = 2 a + 30

2 a = - 16

a = - 16/2

a = - 8

If 'a' is the first term and 'd' is a common difference then the A.P is

a, a + d, a + 2 d, a + 3 d \dots

- 8, - 6, - 4, - 2 \dots

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Question Text	The sum of first $16$ terms of an $A . P$ is $112$ and sum of its next fourteen terms is $518$ . Find the $A . P$ .
Answer Type	Text solution:1
Upvotes	164