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Question
If the sum of first 7 terms of an A.P. is 49 and that of its first 17 terms is 289, find the sum of first n terms of the A.P.
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Solution
Let the first term and the common difference of the given A.P. be a and d, respectively.
Sum of the first 7 terms, S7 = 49
We know
`S = n/2[2a + (n - 1)d]`
⇒ `7/2(2a + 6d) = 49`
⇒ `7/2 xx 2(a + 3d) = 49`
⇒ a + 3d = 7 ...(1)
Sum of the first 17 terms, S17 = 289
⇒ `17/2(2a + 16d) = 289`
⇒ `17/2 xx 2(a + 8d) = 289`
⇒ a + 8d = `289/17`
⇒ a + 8d = 17 ...(2)
Subtracting (2) from (1), we get
5d = 10
d = `5/10`
⇒ d = 2
Substituting the value of d in (1), we get
a = 1
Now,
sum of the first n terms is given by
`S_n = n/2[2a + (n - 1)d]`
= `n/2[2 xx 1 + (n - 1) xx 2]`
= `n/2 [2 + 2n - 2]`
= `n/2 [2n]`
= n2
Therefore, the sum of the first n terms of the A.P. is n2.
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