Question

Find Sn for the arithmetic series 4+7+10 + … and determine the value of n for which the series has sum 175.

Answer 1

`S_(n) = (n)/(2)[3n + 5]`

n = 10

Explanation:

The given arithmetic series is 4 + 7 + 10 + .......... up to n terms.

Now, we know that the sum of first n terms of an A.P. with first term a and the common difference d is given by

`S_(n) = (n)/(2)[2a + (n - 1)d]`

So, in our case the first term a = 4 and the common difference is d = 3, hence the sum of first n terms will be

`S_(n) = (n)/(2)[2* 4 + (n - 1)* 3] = (n)/(2)[3n + 5]` (Answer)

Now, given
`S_(n) = 175` and we have to find the value of n.

So,
`(n)/(2)[3n + 5] = 175`

⇒ 3n² + 5n = 350

⇒ 3n² + 5n - 350 = 0

⇒ 3n² + 35n - 30n - 350 = 0

⇒ (n + 35)(3n - 30) = 0

⇒ n = 10 {Since n can not be negative} (Answer)