Question

The sum of the first five terms of an arithmetic series is 85. The sum of the first six terms is 123. What are the first four terms of the series?

Answer 1

The first four terms are `3, 10, 17, 24`.

Explanation:

If `s_6 = 123` and `s_5 = 85`, then the 6th term must have value `123 - 85 = 38`.

We now write an equation.

`s_n = n/2(t_1 + t_n)`

`123 = 6/2(t_1 + 38)`

`123 = 3t_1 + 114`

`9 = 3t_1`

`t_1 = 3`

Now, we find the common difference.

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

`38 = 3 + (6 - 1)d`

`35 = 5d`

`d = 7`

The first four terms are therefore:

`=3`

`3 + 7 = 10`

`10 + 7 = 17`

`17 + 7 = 24`

Practice exercises

The sum of the first `19` terms of an arithmetic sequence is `x` and the sum of the first `20` terms is `x + 72`. The first term is `15`.
a) What is the 20th term?

b) Find the value of `x`.

Solutions:

a) `72`
b) `798`

Hopefully this helps!