sequence and series?

[btrfly24]

Here's the actual problem:

Sam's retirement plan gives her a fixed raise of d dollars each year. If her retirement income was $24500 her fifth year and $25700 her 9th year, then what was her income the first year and what will it be her 13th year.

Now, I figured out what all of them are, and that d=300, but I can't write out an actual equation as to how I got it. I just did it the long way. Could someone please help set me in the right direction? Thanks in advance.

 

[pka]

Is this what ypu mean?
\(\displaystyle R_n = 23000 + 300n\)

 

[soroban]

Hello, btrfly24!

Do you know anything about Arithmetic Sequences?

If the first term is \(\displaystyle a\) and the common difference is \(\displaystyle d\):

. . the \(\displaystyle n^{th}\) term is: \(\displaystyle \:a_n\;=\;a\,+\,(n\,-\,1)d\)

Sam's retirement plan gives her a fixed raise of \(\displaystyle d\) dollars each year.
If her retirement income was $24500 her fifth year and $25700 her 9th year,
then what was her income the first year and what will it be her 13th year?

She got $24,500 in year 5: \(\displaystyle \:a_5\;=\;a\,+\,4d\;=\;24,500\;\) [1]

She got $25,700 in year 9: \(\displaystyle \:a_9\;=\;a\,+\,8d\;=\;25,700\;\) [2]

Subtract [1] from [2]: \(\displaystyle \:4d\:=\:1200\;\;\Rightarrow\;\;\fbox{d\:=\:300}\)

Substitute into [1]: \(\displaystyle \:a\,+\,8(300)\:=\:24,500\;\;\Rightarrow\;\;\fbox{a\:=\:23,300}\)

Hence, she got \(\displaystyle \fbox{\$23,300}\) the first year.


In her 13th year, she got: \(\displaystyle \,a_{13}\;=\;23,300\,+\,12(300)\;=\;\fbox{\$26,900}\)

 

[btrfly24]

thanks to both of you for your help. I do know about the sequences, but it was about three in the morning and I was having a block I guess. Thanks for breaking it down for me.