2. What is the compound interest of $100 for 4 years,

3. What is the compound interest of $630 for 4 years, at 5 per cent. ? Ans. $135.769.

By carefully reviewing the above manner in which compound interest is computed, we discover that the successive amounts, which are considered as new principals, form the terms of a geometrical series, whose first term is the original principal; the ratio is the amount of $1 for one year, at the given rate per cent.; the number of terms is equal to the number of years, plus one.

From this we learn that finding the amount of a given principal, for a given number of years, at a given rate per cent., consists in finding the last term of a geometrical progression, when the first term, the ratio, and the number of terms are given.

Thus, the amount of $1 for one year, at 3 per cent., is $103; for two years, it is $(103); for three years, it is $(103)3; for four years, it is $(103); and in general, for any number of years, it is found by raising 1.03 to a power denoted by the number of years.

TABLE,

SHOWING the amount of $1, or £1, for any number of years, not exceeding 30, at 3, 4, 5, and 6 per cent., at compound interest, the interest being compounded yearly.

We will now solve the following questions by means of the preceding table.

4. What is the amount of $790 for 13 years, at 6 per cent.?

From our table, we find the amount of $1 for 13 years, at 6 per cent., to be $2132928; this, multiplied by 790, the number of dollars in the principal, gives $1685-013 for the amount required.

5. What is the compound interest of $49, for 20 years, at 5 per cent.?

In this example, we find, from the table, that the amount of $1 for 20 years, at 5 per cent., is $2·653298; which, multiplied by 49, gives $130 012 for the amount of $49, from which, if we subtract $49, we get $81.012 for the compound interest required.

6. What is the compound interest of $100 for 17 years, at 6 per cent.? Ans. $169.277. 7. What is the compound interest of $375 for 20 years, at 6 per cent.? Ans. $827.676. 8. What is the amount of $875 for 12 years, at 6 per cent., compound interest? Ans. $1760672. 9. What is the amount of $625 for 18 years, at 5 per cent., compound interest? Ans. $1504.137. 10. What is the amount of $379 for 30 years, at 3 per cent., compound interest? Ans. $919-932.

NOTE. When the interest is compounded half-yearly, we must take the amount of $1 for half a year, and raise it to a power denoted by the number of half-years in the whole time; this power, multiplied by the principal, will give the amount. We must proceed in a similar way for any other aliquot part of a year. Or, in such cases, we may make use of our table, as in the work of next question.

11. What is the amount of $100 for 3 years, at 6 per

cent. per annum, when the interest is added at the end 6 months?

of every

In this example, we change the 6 per cent. to 3 per cent., and the 3 years to 6 years; we then find the tabular number to be $1.194052; which, multiplied by 100, gives $119-405 for the amount required.

12. What will £600 amount to in 6 years, at 8 per cent., compound interest, supposing the interest to be receivable half-yearly? Ans. £960 12 s. 4 d.

13. What will $890 amount to in 5 years and 4 months, at 9 per cent. per annum, compound interest, the interest being added at the end of every 4 months? Ans. $1428 189.

14. What will $3705 amount to in 3 years and 3 months, at 12 per cent. per annum, compound interest, the interest being added at the end of every 3 months? Ans. $5440 918.

15. What will $378 amount to in 7 years and 6 months, at 8 per cent. per annum, the interest being compounded half-yearly? Ans. $680.757. 16. What will $1000 amount to in 15 years, at 8 per cent. per annum, the interest being compounded halfyearly? Ans. $3243.398.

COMPOUND DISCOUNT.

69. COMPOUND DISCOUNT is an allowance made for the payment of money before it is due, on the supposition that the money draws compound interest.

The present worth of a debt payable at some future period, without interest, is such a sum as being put out

at compound interest, will, in the given time, at the given rate per cent., amount to the debt.

Hence, the finding the present worth resolves itself into the following :

Given the amount at compound interest, the time, and the rate per cent., to find the principal.

Under compound interest, it was shown that the amount was equal to the number of dollars in the principal, multiplied by the amount of $1 for one year, raised to a power whose exponent is the number of years. Hence, we have the following rule to find the principal, or present worth:

RULE.

Divide the given amount by the amount of $1 for 1 year, raised to a power whose exponent is equal to the number of years.

EXAMPLES.

1. What is the present worth of $1685, due 13 years hence, allowing discount according to 6 per cent., compound interest?

From the table under Art. 68, we find that the amount of $1 for 1 year, at 6 per cent., raised to the 13th power, or, what is the same, the number for 13 years is $2132928; .. dividing $1685 by 2132928, gives $789-994 for the present worth required.

The present worth of $1 for one year, at 3 per cent., is for two years, it is (T); for three years, it is (1); for four years, it is (1)*; and in general, the present worth of $1, at compound interest, is the reciprocal of the amount of $1, at compound interest, for the same time and same rate per cent.