Question

How do you write a rule for the nth term of the arithmetic sequence `a_11=29, a_20=101`?

Answer 1

`a_n=8n-59`

Explanation:

Let set up two equation base on the information that given
`a_11=29 n=11` , `29=a_1+(11-1)d` , `29=a_1+10d`

`a_20=101`, `101=a_1+(20-1)d` , `101=a_1+19d`

Now we got:
`101=a_1+19d`
`29=a_1+10d`
We can solve this by using elimination method
So we need to multiply either equation to isolate `a_1`
`101=a_1+19d`
`-29=-a_1-10d`
Add vertically
`72=9d` We can find d (common ratio) by divide 9 on both sides
`d=8`
Now substitute d into either equation that we set up to find `a_1`
`29=a_1+10(8)`
`29=a_1+80`
`a_1=-51`

The rule for an Arithmetic Sequence is `a_n=a_1+(n-1)d`
`d=8` `a_1=-51`
`a_n=-51+(n-1)8`
`a_n=-51+8n-8`
`a_n=8n-59`