solve;
FV=7,000;i=0.03;PMT=$400;n=?
To solve the problem, we need to use the formula for the future value of an ordinary annuity:
FV = PMT * [(1 - (1 + i)^(-n)) / i]
Where:
FV is the future value ($7,000 in this case)
PMT is the periodic payment ($400 in this case)
i is the interest rate per period (0.03 or 3% in this case)
n is the number of periods (unknown in this case)
We need to rearrange the formula to solve for n. Here's the step-by-step process:
1. Divide both sides of the equation by PMT:
FV / PMT = (1 - (1 + i)^(-n)) / i
2. Multiply both sides of the equation by i:
i * (FV / PMT) = 1 - (1 + i)^(-n)
3. Subtract (1 - (1 + i)^(-n)) from both sides of the equation:
i * (FV / PMT) - 1 = -(1 + i)^(-n)
4. Multiply both sides of the equation by -1:
1 - i * (FV / PMT) = (1 + i)^(-n)
5. Raise both sides of the equation to the power of -1:
(1 - i * (FV / PMT))^(-1) = (1 + i)^n
Now, we have isolated n on the right side of the equation. We can take the natural log of both sides to solve for n:
ln((1 - i * (FV / PMT))^(-1)) = ln((1 + i)^n)
By applying the logarithmic property, we can simplify the equation further:
-ln(1 - i * (FV / PMT)) = n * ln(1 + i)
Finally, we can solve for n by plugging in the values and using a calculator:
n = -ln(1 - (0.03 * (7000 / 400))) / ln(1 + 0.03)
n ≈ 11.48
Therefore, the value of n is approximately 11.48.