Question

Example:3 The 1st term of an arithmetic sequence is 200 and the common nunmber is -10 What is the formula an? What is the 20th term?

Answer 1

`a_n = 200+(n-1)*(-10) = 200-10(n-1)`

`a_19 = 10`

Explanation:

An arithmetic sequence starts with an initial value and adds the same constant with every iteration. The terms are thus written like this:

`a_0=x_0`
`a_1=x_0+r`
`a_2 = a_1 + r = (x_0+r)+r=x_0+2r`
`a_3 = a_2 + r = (x_0+2r)+r=x_0+3r`
`...`
`a_n = a_{n-1} + r = (x_0+(n-1)r)+r=x_0+nr`

In your case, the starting number `x_0` is `200`, and the common number that we add every time is `-10`

This means that the formula for the generic term is

`a_n = 200+(n-1)*(-10) = 200-10(n-1)`

To find the `20`th term, just plug `n=19` in the generic equation. In fact, since we're starting from `a_0`, the indices are shifted so that `a_0` is the first term, `a_1` is the second, and so on. We get

`a_19 = 200-10*19 = 200-190=10`