Answer:
1. n = 40
2.
Step-by-step explanation:
The ordinary annuity formula can be written as ...
PV = PMT(1 -(1+r)^-n)/r
where PMT is the payment per period, r is the interest rate per period, and n is the number of periods.
This formula can be solved explicitly for n, but not for r. Iterative or other methods can be used to find r.
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1. Filling in the given information, we have ...
15000 = 650(1 -1.03^-n)/0.03
450/650 = 1 - 1.03^-n . . . . . divide by the coefficient of the stuff in parens
1.03^-n = 4/13 . . . . . . . . . . . solve for the exponential term
-n·log(1.03) = log(4/13) . . . . take logarithms
n = log(13/4)/log(1.03) ≈ 39.87 . . . . . solve for n
n ≈ 40
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2. We can rewrite the annuity formula to make it be a function of i that is zero at the desired value of i.
f(i) = PV -PMT(1 -(1+i)^-n)/i
If we want i as a percentage, then we can replace i with i/100 and fill in the given values to get ...
f(i) = 9000 -500(1 -(1 +i/100)^-35)/(i/100)
f(i) = 1000(9 -50(1 -(1 +i/100)^-35)/i) . . . . multiply the fraction by 100/100
Since we're seeking a value of f(r) that is zero, we can eliminate the factor of 1000.
f(i) = 9 -50(1 - (1+i/100)^-35)/i
The attached graph shows the solution to f(i)=0 is near i=4.27%. As a decimal rounded to 3 decimal places, this is ...
i ≈ 0.043
