Question

Given the first term and the common difference of an arithmetic sequence how do you find the first five terms and explicit formula: a1 = 3/5, d= -1/3?

Answer 1

`a_n = a_1 + (n-1)d`
`A.P(9/15,4/15,-1/15,-6/15,-11/15)`

Explanation:

The concept of an arithmetic sequence is that, from the first term, every term after has a common difference between, so

`a_n - a_(n-1) = a_(n-1) - a_(n-2) = ... = a_3 - a_2 = a_2 - a_1 = d`

Which means, that we can rewrite `a_2` as `a_1 + d`, and we can also rewrite `a_3` as `a_2+d` or `a_1 + 2d` and so on, until we start seeing a pattern. `a_n = a_1 + (n-1)d`, because we can put `a_(n-1)` in terms of `a_(n-2)` and that in terms of `a_(n-3)` and so on down to `a_1`, doing a total of `(n-1)` rewrites.

As for the terms, it's just a matter of evaluating in your matter of choice.
`a_1 = 3/5 = 9/15`
`a_2 = 3/5 - 1/3 = 9/15 - 5/15 = 4/15`
`a_3 = 3/5 -2/3 = 9/15 - 10/15 = -1/15`
`a_4 = 3/5 - 3/3 = 9/15 - 15/15 = -6/15`
`a_5 = 3/5 - 4/3 = 9/15 - 20/15 = -11/15`