Question

Select the correct answer from each drop-down menu.

Nathan has 80 stamps in his collection. He adds 1 stamp to it today. Each day he plans to add twice the number of stamps as the previous day. If he keeps adding stamps at this rate for n days, which recursive function represents the number of stamps he has on any day in the future?
The recursive function is , starting at .

Answer 1

`d_(n+1)=d_n\,+\,2^n`

with
`d_1=81`

Explanation:

We can study what happens day by day starting with day one (1) to understand how the recursive relation can be constructed:

Day # Number of stamps What is added relative to previous day

1 d1 = 80 + 1 = 81

2 d2 = 81 + 2 = 83 d2 = d1 + 2

3 d3 = 83 + 4 = 87 d3 = d2 +
`2^2`

4 d4 = 87 + 8 = 95 d4 = d3 +
`2^3`

5 d5 = 95 + 16 = 111 d5 = d4 +
`2^4`

6 d6 = 111 + 32 = 143 d6 = d5 +
`2^2`

(n+1) d(n+1) = dn +
`2^n`