Question

Find the sum of the following arithmetic series 7+13+19+25+...+85

Answer 1

Explanation:

Given Arithmetic series is:

7+13+19+25+...+ 85

Here,

First term a = 7

Common Difference d = 13 - 7 = 6

last term
`t_n= 85`

First let us find the number of terms in given series.

`t_n= a + (n-1)d\\</p><p>\therefore 85 = 7+ (n-1)* 6\\</p><p>\therefore 85 = 7+ 6n-6\\</p><p>\therefore 85 = 1+ 6n\\</p><p>\therefore 6n = 85 - 1</p><p>\therefore 6n = 84</p><p>\therefore n = (84)/(6)\\</p><p>\therefore n = 14\\</p><p>`

Hence, given series has total 14 terms.

Sum of n terms of an Arithmetic series is given as:

`S_n = (n)/(2) (a + t_n) \\ \\ \therefore \: S_(14)= (14)/(2) * \: (7 + 85) \\ \\ \therefore \: S_(14)= 7 * \: 92 \\ \\ \huge \red{ \boxed{\therefore \: S_(14)= 644 }}\\`