Question

Find the sum sn of the arithmetic sequence a7＝14/3 d＝-4/3 n＝15

Answer 1

`S_(15)= 50`

Explanation:

Given

`a_7 = (14)/(3)`

`d = -(4)/(3)`

`n = 15`

Required

The sum of n terms

First, we calculate the first term using:

`a_n = a + (n - 1)d`

Let
`n = 7`

So, we have:

`a_7 = a + (7 - 1)d`

`a_7 = a + 6d`

Substitute
`a_7 = (14)/(3)` and
`d = -(4)/(3)`

`(14)/(3) = a + 6*(-4)/(3)`

`(14)/(3) = a -8`

Collect like terms

`a =(14)/(3) +8`

Take LCM and solve

`a =(14+24)/(3)`

`a =(38)/(3)`

The sum of n terms is then calculated as:

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

Where:
`n = 15`

So, we have:

`S_n = (15)/(2)(2*(38)/(3) + (15 - 1)*(-4)/(3))`

`S_n = (15)/(2)(2*(38)/(3) + 14 *(-4)/(3))`

`S_n = (15)/(2)(2*(38)/(3) - 14 *(4)/(3))`

`S_n = (15)/(2)((2*38)/(3) - (14 *4)/(3))`

Take LCM

`S_n = (15)/(2)((2*38-14 *4)/(3))`

`S_n = (15)/(2)((20)/(3))`

Open bracket

`S_n = (15*20)/(2*3)`

`S_n = (300)/(6)`

`S_n = 50`

Hence,

`S_(15)= 50`