Question

In an arithmetic series t6 = −4, t10 = −12. Find S10.

Answer 1

`S_(10)` = - 30

Explanation:

For a given arithmetic sequence the n th term formula is

`t_(n)` =
`t_(1)` + (n - 1)d

where d is the common difference and
`t_(1)` the first term

We have to find d and
`t_(1)`

from the given information we can write 2 equations and solve for d and
`t_(1)`

`t_(6)` =
`t_(1)` + 5d = - 4 → (1)

`t_(10)` =
`t_(1)` + 9d = - 12 → (2)

subtract (1) from (2) term by term

4d = - 8 ⇒ d = - 2

substitute d = - 2 in (1)

`t_(1)` - 10 = - 4 ⇒
`t_(1)` = - 4 + 10 = 6

The sum to n terms of an arithmetic sequence is

`S_(n)` =
`(n)/(2)`[2
`t_(1)` + (n - 1)d ], hence

`S_(10)` = 5[(2 × 6) + (9 × - 2) ] = 5(12 - 18) = 5 × - 6 = - 30