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Sum

The first term of an AP is –5 and the last term is 45. If the sum of the terms of the AP is 120, then find the number of terms and the common difference.

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#### Solution

Let the first term, common difference and the number of terms of an AP are a, d and n respectively.

Given that, first term (a) = −5 and

last term (l) = 45

Sum of the terms of the AP = 120

⇒ S_{n} = 120

We know that, if last term of an AP is known, then

Sum of n terms of an AP is,

`S_n = n/2 (a + 1)`

⇒ 120 = `n/2(-5 + 45)`

⇒ `120 xx 2 = 40 xx n`

⇒ `n = 3 xx 2`

⇒ n = 6

∴ Number of terms of an AP is known

Then the n^{th} term of an AP is,

l = a + (n – 1)d

⇒ 45 = –5 + (6 – 1)d

⇒ 50 = 5d

⇒ d = 10

So, the common difference is 10.

Hence, number of terms and the common difference of an AP are 6 and 10 respectively.

Concept: Sum of First n Terms of an A.P.

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