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PROBLEM V. To find any number of mean proportionals between two given numbers.

RULE. · The two given numbers are the extremes of a series consisting of two more terms than there are means required; hence the ratio will be found by problem 4.

Then the product of the ratio and the less extreme will be one of the means; the product of this mean and the ratio will be another mean; and so on, till all the required means are found.

When only one mean is required, it is the square root of the product of the extremes.

21. Find 3 mean proportionals between 5 and 1280.

Here the series is to consist of five terms, and the extremes are 5 and 1280; hence the ratio is found by the fourth problem to be 4; and by the repeated multiplication of the least term by the ratio, the means are found to be 20, 80, and 320.

22. Find four mean proportionals between 4 and 2401.

23. Find five mean proportionals between the numbers, 279936 and 6.

24. Find a mean proportional between 1 and 2809.

COMPOUND INTEREST BY SERIES. It has been shown in Art. xv, page 107, that compound interest is that which arises from adding the interest to the principal at the end of each year, and taking the amount for a new principal. Now, the several amounts for the several years form a series of continual propor-. tionals; and, to find the amount for any number of years, we may adopt the following

RULE. Find the last term of an increasing series of continual proportionals, whose first term is the principal, whose ratio is the amount of 1 dollar for'l year, and whose number of terms is the number of years-plus 1. The last term is the required amount. See Problem 1st..

In the examples under this rule, no more than six decimal places need be included.

25. What is the amount of $100, at 6 per cent. compound interest, for 4 years ?

26. What is the amount of $75, at 5 per cent. compound interest, for 9 years ?

27. What is the amount of $ 294, at 4 per cent. compound interest, for 7 years ?

28. Find the amount of $18.25, at 7 per cent. compound interest, for 12 years.

29. Find the amount of $751.30, at 5 per cent. compound interest, for 8 years.

30. Find the amount of $4798, at 6 per cent. compound interest, for 12 years.

31. What is the amount of $ 5.14, at 7 percent. compound interest, for 16 years ? What is the interest? - 32. What is the compound interest of $ 1000 for 20 years, at 6 per cent. ? What is the amount ?

• COMPOUND DISCOUNT. Discount corresponding to simple interest has already been treated, in Art. XVI; but discount corresponding to compound interest is now to be computed.

On the supposition that money can be let out at compound interest, the present worth of a debt, payable at a future period without interest, is that principal, which, at compound interest, would give an amount equal to the debt, at the period when the debt is payable.

RULE. Find the last term of a decreasing series of continual proportionals, whose first term is the debt, whose ratio is the amount of 1 dollar for 1 year, and whose number of terms is the number of years plus 1. The last term is the present worth. See Problem 1st.

33. What principal, at 10 per cent. compound interest, will amount, in 4 years, to $ 8.7846 ?

34. What is the present worth of $ 68.40, payable 11 years hence; allowing discount according to 5 per cent. compound interest ?

35. What is the present worth of $ 350, payable in 5 years; allowing discount at the rate of 6 per cent. compound interest?

36. What is the present worth of $3525, due in 3 years; discount being allowed as in the last example ?

37. How much must be advanced to discharge a debt of $ 700, due in 8 years; discounting at the rate of 5 per cent. compound interest?

38. What is the present worth of $1000, due in 20 years; discounting at the rate of 6 per cent. compound interest? How much is the discount?

XXXIV.

ANNUITIES. An ANNUITY is a fixed sum of money payable periodically, for a certain length of time, or during the life of some person, or for ever.

Although the term annuity, in its proper sense, applies only to annual payments, yet payments which are made semiannually, quarterly, monthly, &c., are also called annuities.

Pensions, salaries, and rents, come under the head of annuities. Annuities may, however, be purchased by the present payment of a sum of money. The party selling annuities, is usually an incorporated trust company, instituted and regulated upon principles similar to those of an insurance company. The company has an office, called an annuity office, where all its business is transacted.

The present worth of an annuity which is to continue for ever, is that sum of money, which would yield an interest equal to the annuity. But the present worth of an annuity which is to terminate, is a sum, which, being put on compound interest, would, at the termination of the annuity, amount to just as much as the payments of the annuity would amount to, provided they should severally be put on compound interest, as they became due.

The sum to be paid for the purchase of a life annuitywhich is the same as its present worth-depends not only upon the rate of interest, but, also upon the probable continuance of the life or lives on which the annuity is granted. In order to bring data of this kind into numbers, the bills of mortality in different places have been examined, and from them, tables have been constructed, which show how many persons, upon an average, out of a certain · number born, are left alive at the end of each year; and

from these tables others have been constructed, showing the expected continuance of human life, at every age, according to probabilities. We shall not, however, treat the subject of life annuities in this work, and would refer readers, who wish to become thoroughly acquainted with its theory, to the writings of Simpson, De Moivre, Bailey, Price, and Milne.

PROBLEM I. To find the amount of an annuity, which has been forborn for a given time.

Before presenting the rule, let us inquire what would be the amount of an annuity of $ 100, forborn 4 years, allowing 5 per cent. compound interest? The last year's payment will, obviously, be $100 without interest; the last but one will be the amount of $100 for 1 year; the last but two will be the amount of $100 for 2 years; and so on: and the sum of the amounts will be the answer. Now the last payment with the amounts for the several years, form a series of continual proportionals. We, therefore, adopt the following

RULE. Find the sum of an increasing series of continual proportionals, whose first term is the annuity, whose ratio is the amount of 1 dollar for 1 year, and whose number of terms is the number of years. This sum is the amount. See Art. XXXII, Problems 1st and 2nd..

1. What is the amount of an annuity of $ 200, which has been forborn 14 years; allowing 6 per cent. interest ?

2. What is the amount of an annuity of $ 50, which has been forborn 20 years; interest being 5 per cent. ?

3. What is the amount of an annual rent of $150, for born 7 years; allowing interest at 5 per cent. ?

4. If the annual rent of a house be in arrears 4 years, what is the amount, allowing 10.per cent. interest ?

5. Suppose a person, who has a salary of $600 a year, payable quarterly, to allow it to remain unpaid for 3 years, how much would be due him; allowing quarterly compound interest at 6 per cent. per annum ?

6. What is due on a pension of $150 a year, payable half-yearly, but forborn 2 years; allowing half-yearly compound interest, at 45 per cent. per annum ?

7. What is due on a pension of $ 300 a year, payable quarterly, but forborn 21 years; allowing quarterly compound interest, at 5 per cent. per annum.

PROBLEM II. To find the present worth of an annuity which is to terminate in a given number of years.

Before giving the rule, let us inquire, what is the present worth of an annuity of $ 100, to continue 4 years, allowing 5 per cent. interest ? The present worth is, obviously, a sum, which, at compound interest, would produce an amount equal to the amount of the annuity. Now we can find the amount of any sum at compound interest, by multiplying the sum by the amount of 1 dollar for a year, as many times as there are years. Hence, to find a sum which will produce a given amount in a given time, we must reverse the process, and divide by the amount of 1 dollar for the time. Applying this principle to the example in question, we find by the preceding rule, that the amount of the annuity is $431. Then, dividing this amount by the amount of 1 dollar for 4 years, we find the present worth to be $ 354.593+

RULE. Find the amount of the annuity as if it were in arrears for the whole time, and divide this amount by the amount of 1 dollar at compound interest for the same time; the quotient will be the present worth.

8. What is the present worth of an annuity of $ 500, to continue 10 years; interest being 6 per cent. ?

9. What is the present worth of an annuity of $ 80, to continue 22 years; interest being 5 per cent. ?

The operations in this rule being tedious, we introduce, upon the next page, a table, showing the present worth of $1 annuity, at 4, 5, 6, and 7 per cent., for every number of years, from 1 to 30. To find the present worth of an annuity by the use of this table, multiply the present worth of 1 dollar for the number of years, by the annuity.

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