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

Answer:

$S_{n} = \frac{n}{2}[3n + 5]$

n = 10

Step-by-step 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

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

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

$S_{n} = \frac{n}{2}[2\times 4 + (n - 1)\times 3] = \frac{n}{2}[3n + 5]$ (Answer)

Now, given $S_{n} = 175$ and we have to find the value of n.

So, $\frac{n}{2}[3n + 5] = 175$

⇒ 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)