Question

What is `a_1` and `d` for an arithmetic sequence with `a_3 = -8` and `a_7=32`?

Answer 1

Different people have different ways of attacking such problems but my method involves systems of equations.

Recall the nth term of an arithmetic sequence is given by `t_n = a + (n - 1)d`. Then:

`32 = a + (7 - 1)d`
`-8 = a + (3 - 1)d`

Solving for `a` and for `d`:

`32 = a + 6d`

`32 - 6d = a`

`-8 = 32 - 6d + 2d`

`-8 - 32 = -4d`

`-40 = -4d`

`10 = d`

We can now use the formula `t_n = a + (n - 1)d` to find `a`

`-8 = a + (3 - 1)10`

`-8= a + 20`

`-8 - 20 = a`

`a = -28`

In summary...

`d = 10`

`a = -28`

Practice exercises:

1. If in an arithmetic sequence the fifth term has a value of `3` and the sixteenth term has a value of `-41`, determine:

a) `d`

b) `t_75`

c) `s_26`

Hopefully this helps, and good luck!