Review question

# How are these recursively defined sequences related? Add to your resource collection Remove from your resource collection Add notes to this resource View your notes for this resource

Ref: R6555

## Suggestion

A list of real numbers $x_1$, $x_2$, $x_3$, … is defined by $x_1 = 1$, $x_2 = 3$ and then for $n\geq3$ by $x_n = 2x_{n−1} −x_{n−2} + 1.$

So, for example, $x_3 = 2x_2 −x_1 +1=2\times3−1+1=6$.

1. A second list of real numbers $y_1$, $y_2$, $y_3$, … is defined by $y_1 = 1$ and $y_n = y_{n−1} + 2n.$

Find, explaining your reasoning, a formula for $y_n$ which holds for $n\geq2$.

Write out the first few terms of $y_n$ – what kind of sequence is this? Are there any familiar patterns in the $n$th term?

What is the approximate value of $\dfrac{x_n}{y_n}$ for large values of $n$?

Could we write $y_n$ in terms of $x_n$? Or $x_n$ in terms of $y_n$?