Question

Linear recurrence relation

Answer 1

True

A linear recurrence relation involving a sequence of numbers $a_n$ is one of the form

$\displaystyle\sum_{k=0}^nc_{n-k}a_{n-k}=c_na_n+c_{n-1}a_{n-1}+\cdots+c_2a_2+c_1a_1=c$

where $c_1,c_2,\ldots,c_n$ and $c$ are any fixed numbers.

The given recurrence can be rearranged as

$a_n=a_{n-1}+2\implies 1\cdot a_n+(-1)\cdot a_{n-1}=2$

A nonlinear recurrence would have a more "exotic" form that cannot be written in the form above. Some example:

$a_n+\dfrac1{a_{n-1}}=1$

$a_na_{n-1}=\pi$

${a_n}^2+\sqrt{a_{n-1}}-\left(\dfrac{a_{n-2}}{\sqrt{a_n}}\right)^{a_{n-3}}=0$