The starting price of a hotem room is $48.00.
After 1 year, this price increases by 5.1%, meaning it's increased by a factor of 1.051 or 1 + 0.051:
$48.00 (1 + 0.051) ≈ $50.45
After another year, this price increases by another 5.1%, so the previous amount is increased by a factor of 1.051, which is equivalent to 1.051² times the initial price:
$50.45 (1 + 0.051) = $48.00 (1 + 0.051)² ≈ $53.02
After another year, another increase:
$53.02 (1 + 0.051) = $48.00 (1 + 0.051)³ ≈ $55.72
And so on. You should start to see where the given formula comes from,
where a = $48.00 is the starting price, r = 0.051 is the rate at which the price increases, and t is number of years.
Then in 32 years, the price of the room will be
$48.00 (1 + 0.051)³² ≈ $235.79