Number sequences: Find nth term for P(1) = MP(0) + C where M and C are constants

[jftp]

I want to work out what the nth term is in a sequence where each new term P(1) = MP(0) + C where M and C are constants. I.e. multiply the previous value by M and add C. It's not an arithmetic or geometric sequence but kind of both and I can't work it out. Any clues?

 

[Dr.Peterson]

I want to work out what the nth term is in a sequence where each new term P(1) = MP(0) + C where M and C are constants. I.e. multiply the previous value by M and add C. It's not an arithmetic or geometric sequence but kind of both and I can't work it out. Any clues?

Please show us what you have tried, so we can help you with the next step. Keep in mind how we work:

Posting Guidelines (Summary)

Welcome to our tutoring boards! :) This page summarizes the main points from our posting guidelines. As our name implies, we provide math help (primarily to students with homework). We do not generally post immediate answers or step-by-step solutions. We don't do your homework. We prefer to...

www.freemathhelp.com

I would start by writing out the first several terms supposing that P(0) = x, M = a, and C = b. You should see a pattern that you can use.

 

[jftp]

Thank you, that has got me a bit further down the road.
For my terms I get

n
1: x
2: mx+x
3: m(mx+c)+c
4: m(m(mx+c)+c)+c
5: m(m(m(mx+c)+c)+c)+c

I've attempted to expand out the brackets and simplify below

n
1: x
2: mx+c
3: m^2x + mc + c
4: m^3x + m^2c + mc + c
5: m^4x + m^3c + m^2c + mc + c

Although this works great for small values of n, I'm not sure I'd want to work out the 100th term. Just thinking about it, it's a lot like working out the balance of an investment. Perhaps I'll have a go at applying the FV = PMT * (1 + r)^n -1 / r formula to it, or am I going off on a completely wrong tangent here?

 

[Dr.Peterson]

I've attempted to expand out the brackets and simplify below

n
1: x
2: mx+c
3: m^2x + mc + c
4: m^3x + m^2c + mc + c
5: m^4x + m^3c + m^2c + mc + c

Although this works great for small values of n, I'm not sure I'd want to work out the 100th term. Just thinking about it, it's a lot like working out the balance of an investment. Perhaps I'll have a go at applying the FV = PMT * (1 + r)^n -1 / r formula to it, or am I going off on a completely wrong tangent here?

Do you not see the pattern, which easily extends for any n?

5: m^4x + (m^3 + m^2 + m + 1)c​

Do you know the formula for a geometric series? It's related to the formula you show, but more directly relevant.