How many annual deposits of $1,500, beginning next year, would you need to make before you had accumulated $30,000, if the money earns 9% per year, compounded annually? Assume the account begins with a $0 balance.
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A. B. C. D. E.C
On the BAII Plus, press 9 I/Y, 0 PV, 1500 PMT, 30000 +/- FV, CPT N. On the HP12C, press 9 i, 0 PV, 1500 PMT, 30000 CHS FV, n. Note that the HP12C will display 12 as the answer.
To calculate the number of annual deposits needed to accumulate $30,000, we can use the formula for the future value of an ordinary annuity:
FV=P×(r(1+r)n−1)
Where: FV = Future value of the annuity P = Annual deposit amount r = Annual interest rate n = Number of deposits
In this case, the annual deposit amount (P) is $1,500, the annual interest rate (r) is 9% (0.09 as a decimal), and we want to accumulate $30,000 (FV). We need to find the value of n, the number of deposits.
Plugging in the values we know, the equation becomes:
30000=1500×(0.09(1+0.09)n−1)
Now, let's solve for n.
0.09(1+0.09)n−1=150030000
(1+0.09)n−1=20
Now, we can solve for n by taking the logarithm of both sides:
log(1+0.09)n=log21
n×log(1+0.09)=log21
Dividing both sides by log(1+0.09):
n=log(1+0.09)log21
Using a calculator, we can find the value of n to be approximately 14.01.
Therefore, the answer is A. 14.01.
You would need to make approximately 14.01 annual deposits of $1,500 to accumulate $30,000 in the account.