1. ## Recurrence Sequences

A sequence of terms $\displaystyle {U_n}$ is defined by $\displaystyle n\geq1$ by the recurrence relation $\displaystyle U_{n+1}=kU_n+2$, where $\displaystyle k$ is a constant. Given that $\displaystyle U_1=3$:

a) Find an expression in terms of $\displaystyle k$ for $\displaystyle U_2$.

Please tell me how to do this .

2. Originally Posted by Viral
A sequence of terms $\displaystyle {U_n}$ is defined by $\displaystyle n\geq1$ by the recurrence relation $\displaystyle U_{n+1}=kU_n+2$, where $\displaystyle k$ is a constant. Given that $\displaystyle U_1=3$:

a) Find an expression in terms of $\displaystyle k$ for $\displaystyle U_2$.

Please tell me how to do this .
Let $\displaystyle n = 1$ in $\displaystyle U_{n+1}=kU_n+2$, so

$\displaystyle U_{2}=k U_1+2 = 3k + 2$.

3. part b says "hence find an expression for $\displaystyle U_3$". The answer is $\displaystyle 3k^2+2k+2$ but I have no idea how to get this answer. Do you know of any notes on the internet with recurrence sequences?

4. Originally Posted by Viral
part b says "hence find an expression for $\displaystyle U_3$". The answer is $\displaystyle 3k^2+2k+2$ but I have no idea how to get this answer. Do you know of any notes on the internet with recurrence sequences?
Since $\displaystyle U_{n+1} = k U_n +2$ then substitute $\displaystyle n = 2$ so

$\displaystyle U_3 = kU_2 + 2$ but you already know $\displaystyle U_2 = 3k +2$ from the first part.

You might try googleing it on the web. Like I did to get this site

Difference Equation Tutorial

Try the word "difference equation".