455. To find the present value of an annuity to continue for a given number of years, allowing compound interest.
Let A be the annuity, R the amount of $1 in one year, n the number of years, V the required present value.
The present vahie of A due in 1 year is AR^-1; the present value of A due in 2 years is AR^-2; the present value of A due in 3 years is AE^-1; and so on. [Art. 452.]
Now V is the sum of the present values of the different payments; V= AR^-1 + AR^-2 + AR^-3 + to n terms = AR^-1 1-R^-n 1 -R^-1 = A 1-R^n R-1
Note. This result may also be obtained by dividing the value of M, given in Art. 454, by R^n [Art. 451.]
Cor. If we make n infinite we obtain for the present value a perpetual annuity V = A R-1 = A r
EXAMPLES XXXIX.
1. If in the year 1600 a sum of $1000 had been left to accumulate for 300 years, find its amount in the year 1900, reckoning compound interest at 4 per cent per annum. Given log 104 = 2.0170333 and log 12885.5 = 4.10999.
2. Find in how many years a sum of money will amount to one hundred times its value at 5 1 2 per cent per annum compound interest. Given log 1055 = 3.023.
3. Find the present value of $6000 due in 20 years, allowing compound interest at 8 per cent per annum. Given log2 = .30103, log3 = .47712, and log 12875 = 4.10975.
4. Find the amount of an annuity of $100 in 15 years, allowing compound interest at 4 per cent per annum. Given log 1.04 = .01703, and log 180075 = 5.25545.
5. What is the present value of an annuity of $1000 due in 30 years, allowing compound interest at 5 per cent per annum ?
