Question

Given the nth term for each arithmetic sequence, how to find the common difference and write out the first four terms here? (1) `a_n=2n+7` (2) `a_n=3n-2`

Answer 1

See below.

Explanation:

Since every couple of consecutive terms in an arithmetic sequence differ by a common difference, we can subtract any two consecutive terms to find out how distant they are from each other.

So, in the first case, let's consider the `n^"th"` and the `n+1^"th"` term:

`a_n=2n+7,\qquad a_{n+1}=2(n+1)+7 = 2n+2+7=2n+9`

and subtract them:

`a_{n+1}-a_n=2n+9-(2n+7)=cancel(2n)+9-cancel(2n)-7=2`

As for the first four terms, it depends if you consider the first term to be associated with `n=0` or `n=1`. I usually go for the first choice. With this assumption, the first four terms are

`a_0 = 2*0 + 7 = 0+7 = 7`
`a_1 = 2*1 + 7 = 2+7 = 9`
`a_2 = 2*2 + 7 = 4+7 = 11`
`a_3 = 2*3 + 7 = 6+7 = 13`

In other words, you simply have to plug the value of `n` in the expression for the general term.

The second sequence behaves exactly in the same way.