Answer
$a_n = a_{n-1}+ a_{n-2} + 2 \times a_{n-5} + 2 \times a_{n-10} + a_{n-20} + a_{n-50} + a_{n-100}$
Work Step by Step
For $n\geq100$ ,
We have to find $a_n$, where $a_n$ denotes the number of ways the payment can be made.
We could pay a 1 dollar coin and pay the rest in $a_{n-1}$ ways or,
we could pay a 2 dollar coin and pay the rest in $a_{n-2}$ ways or,
we could pay a 5 dollar coin and pay the rest in $a_{n-5}$ ways.or,
we could pay a 5 dollar bill and pay the rest in $a_{n-5}$ ways or,
we could pay a 10 dollar coin and pay the rest in $a_{n-10}$ ways or,
we could pay a 10 dollar bill and pay the rest in $a_{n-10}$ ways or,
we could pay a 20 dollar bill and pay the rest in $a_{n-20}$ ways or,
we could pay a 50 dollar bill and pay the rest in $a_{n-50}$ ways or,
we could pay a 100 dollar bill and pay the rest in $a_{n-100}$ ways.
This all totals up to the number of ways we can pay n dollar.
Therefore, $a_n = a_{n-1}+ a_{n-2} +a_{n-5}+ a_{n-5} + a_{n-10}+ a_{n-10} + a_{n-20} + a_{n-50} + a_{n-100}$
Or $a_n = a_{n-1}+ a_{n-2} + 2 \times a_{n-5} + 2 \times a_{n-10} + a_{n-20} + a_{n-50} + a_{n-100}$
