Respuesta :
The recurrence relation for the amount in the account at the end of n years is F(n) = 1.09 × F(n-1), The money will the account contains after 100 years is $5,529,040.80
a) Let F(0) represent the $1,000 initial deposit and F(n) represent the amount in the bank after n years.
F(n) = F(n-1) +0.09 × F(n-1) [ where 9% = 0.09]
F(n) = 1.09 × F(n-1)-------------eq(1)
b) using above formula
F(n) = 1.09 × F(n-1)
F(n) = 1.09 ×( 1.09 × F(n-2)) = [tex]1.09^2[/tex] ×F(n-2)------(replacing n by n-1 in eq (1))
F(n) = [tex]1.09^2[/tex]×( 1.09 × F(n-3))= [tex]1.09^3[/tex] ×F(n-3)---(replacing n by n-2 in eq (1))
∴On generalizing the above expansion, we get
F(n) = [tex]1.09^n[/tex] × F(n-n) = [tex]1.09^n[/tex] × F(0)
F(n) = [tex]1.09^n[/tex] × 1000---------- eq(2)
c) money in the account when n = 100 years have passed.
using equation (2), we get
F(100) = [tex]1.09^1^0^0[/tex] × 1000 = 5,529,040.80
- Recurrence relations are equations that depict sequences based on rules. It assists in determining the following term, which is dependent upon the preceding term. In a given series, we can quickly ascertain the following term if we know the previous term. We can locate the set of new terms now that a common pattern has emerged. This holds true for geometric and mathematical sequences as well.
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