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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 .

Respuesta :

mberisso

Answer:

[tex]d_{n+1}=d_n\,+\,2^n[/tex]

with [tex]d_1=81[/tex]

Step-by-step 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 + [tex]2^2[/tex]

4              d4 = 87 + 8 = 95           d4 = d3 + [tex]2^3[/tex]

5              d5 = 95 + 16 = 111          d5 = d4 + [tex]2^4[/tex]

6              d6 = 111 + 32 = 143        d6 = d5 + [tex]2^2[/tex]

(n+1)                                               d(n+1) = dn + [tex]2^n[/tex]

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