# Given 10, 17, 24, 31, the nth term is given by 7n+3. How you know?

##### 1 Answer

By realizing that

An **arithmetic sequence** is one where some constant difference between numbers in a sequence gives you a pattern to get to the next unlisted number in the sequence.

In this case, the difference is

So,

So, the arithmetic sequence would be generated as:

#color(green)(a_0) = 3 + 7(0) = color(green)(3)#

#color(green)(a_(1)) = a_0 + 7(1) = color(green)(10)#

#color(green)(a_(2)) = a_(1) + 7(1) = a_0 + 7(1) + 7(1) = color(green)(17)#

#color(green)(a_(3)) = a_(2) + 7(1) = a_1 + 7(1) + 7(1)#

#= a_0 + 7(1) + 7(1) + 7(1) = color(green)(24)#

and so on. Each term has **recursive definition**:

#color(blue)(a_n = a_(n-1) + 7)#

or the **overall definition**:

#color(blue)(a_n = 3 + 7n)#