Question

What is the nth term rule of the linear sequence below?

7
,
9
,
11
,
13
,
15
,
.
.
.

Answer 1

`a_n=5+2n`

Explanation:

Each term is 2 more than the previous term, so the common difference is 2.

Let
`a_n` be the nth term of the sequence. Then, the first term is
`a_1=7`.

The second term is
`a_2=7+2=9`, and the third term is
`a_3=7+2+2=11`.

Therefore, every term can be written as

`a_n=7+\underbrace{2+2\cdots +2}_{n-1~\text{times}}`.

Since repeated addition is multiplication, this can be written as

`a_n=7+2(n-1),`

which is the nth term rule of the linear sequence.

This could also be expanded to
`\boxed{a_n=5+2n.}`