Question

Indicate the general rule for the arithmetic sequence with a3 = -12 and a8 = -37

Answer 1

The general rule is
`a_n = -2 - 5(n-1)`

Explanation:

Arithmetic sequence:

In an arithmetic sequence, the difference between consecutive terms is always the same, and this difference is called common difference.

The general rule of an arithmetic sequence is given by:

`a_(n) = a_1 + (n-1)d`

In which
`a_1` is the first term and d is the common difference.

We can also find the nth term as a function of a term m, using:

`a_n = a_m + (n-m)d`

a3 = -12 and a8 = -37

First we find the common difference. So

`a_n = a_m + (n-m)d`

`a_8 = a_3 + (8-3)d`

`-37 = -12 + 5d`

`5d = -25`

`d = -(25)/(5)`

`d = -5`

So

`a_n = a_1 - 5(n-1)`

Finding the first term:

`a_n = a_1 - 5(n-1)`

Since
`a_3 = -12`

`a_n = a_1 - 5(n-1)`

`a_3 = a_1 - 5(3-1)`

`a_1 = a_3 + 10 = -12 + 10 = -2`

So the general rule is:

`a_n = a_1 - 5(n-1)`

`a_n = -2 - 5(n-1)`