RULE. Raise the ratio to a power indicated by the number of terms, and subtract 1 from the result; then multiply this remainder by the first term, and divide the product by the ratio less 1.

EXAMPLES FOR PRACTICE.

1. The first term is 7, the ratio 3, and the number of terms 4; what is the sum of the series?

Ans. 280. 2. The first term is 375, the ratio, and the number of terms 4; what is the sum of the series?

3. The first term is 175, the ratio 1.06, and the number of terms 5; what is the sum of the series? Ans. 986.49+.

PROBLEM VI.

716. Given, the extremes and the sum of the series, to find the ratio.

If we take the geometrical progression, 2, 6, 18, 54, 162, in which the ratio is 3, and remove the first term and the last term, successively, and then compare the results, we have

6+18+54 + 162 :
2+6+18+ 54:

sum of the series minus the first term.

sum of the series minus the last term.

Now, since every term in the first line is 3 times the corresponding term in the second line, the sum of the terms in the first line must be 3 times the sum of the terms in the second line. Hence the

RULE. Divide the sum of the series minus the first term, by the sum of the series minus the last term.

EXAMPLES FOR PRACTICE.

1. The extremes are 2 and 686, and the sum of the series is 800; what is the ratio?

Ans. 7.

2. The extremes are and 64, and the sum of the series is 1273; what is the ratio?

3. If the sum of an infinite series be 41, and the greater extreme 3, what is the ratio?

Ans. 1.

717. Every other problem in Geometrical Progression, that admits of an arithmetical solution, may be solved either by reversing or combining some of the problems already given.

COMPOUND INTEREST BY GEOMETRICAL PROGRESSION.

718. We have seen (550) that if any sum at compound interest be multiplied by the amount of $1 for the given interval, the product will be the amount of the given sum or principal at the end of the first interval; and that this amount constitutes a new principal for the second interval, and so on for a third, fourth, other interval.

or any

Hence,

A question in compound interest constitutes a geometrical progression, whose first term is the principal; the common multiplier or ratio is one plus the rate per cent. for one interval; the number of terms is equal to the number of intervals +1; and the last term is the amount of the given principal for the given time. All the usual cases of compound interest and discount computed at compound interest, can therefore be solved by the rules for geometrical progression. For example,

Find the amount of $250 for 4 years, at 6 % compound interest.

$250 x 1.064

OPERATION.

$250 × 1.262477 = $316.21925.

ANALYSIS. Here we have $250 the first term, 1.06 the ratio, and 5 the number of terms, to find the last term. Then by 711 we find the last term, which is the amount required.

EXAMPLES FOR PRACTICE.

1. What is the amount of $350 in 4 years, at 6 % per annum compound interest ? Ans. $441.86. 2. Of what principal is $150 the compound interest for 2 years, at 7 %?

3. What sum at 6 % compound interest, will amount to $1000 in 3 years

s? Ans. $839.62. 4. In how many years will $40 amount to $53.24, at 10% compound interest? Ans. 3 years.

5. At what rate per cent. compound interest will any sum double itself in 8 years? Ans. 9.05%. 6. What is the present worth of $322.51, at 5 % compound interest, due 24 years hence?

Ans. $100.

ANNUITIES.

719. An Annuity is literally a sum of

money

which is pay

able annually. The term is, however, applied to a sum which is payable at any equal intervals, as monthly, quarterly, semi-annually, etc.

NOTE. The term, interval, will be used to denote the time between payments. Annuities are of three kinds: Certain, Contingent, and Perpetual.

720. A Certain Annuity is one whose period of continuance is definite or fixed.

721. A Contingent Annuity is one whose time of commencement, or ending, or both, is uncertain; and hence the period of its continuance is uncertain.

722. A Perpetual Annuity or Perpetuity is one which continues forever.

723. Each of these kinds is subject, in reference to its commencement, to the three following conditions:

1st. It may be deferred, i. e., it is not to be entered upon until after a certain period of time.

2d. It may be reversionary, i. e., it is not to be entered upon until after the death of a certain person, or the occurrence of some

certain event.

3d. It may be in posscssion, i. e., it is to be entered upon at

once.

724. An Annuity in Arrears or Forborne is one on which the payments were not made when due. Interest is to be reckoned on each payment of an annuity in arrears, from its maturity, the same as on any other debt.

ANNUITIES AT SIMPLE INTEREST.

725. In reference to an annuity at simple interest, we observe: I. The first payment becomes due at the end of the first interval, and hence will bear interest until the annuity is settled.

II. The second payment becomes due at the end of the second interval, and hence will bear interest for one interval less than the first payment.

III. The third payment will bear interest for one interval less than the second; and so on to any number of terms. Hence,

IV. All the payments being settled at one time, each will be less than the preceding, by the interest on the annuity for one interval. Therefore, they will constitute a descending arithmetical progression, whose first term is the annuity plus its interest for as many intervals less one as intervene between the commencement and settlement of the annuity; the common difference is the interest on the annuity for one interval; the number of terms is the number of intervals between the commencement and settlement of the annuity; and the last term is the annuity itself.

726. The rules in Arithmetical Progression will solve all problems in annuities at simple interest.

EXAMPLES FOR PRACTICE.

1. A man works for a farmer one year and six months, at $20 per month, payable monthly; and these wages remain unpaid until the expiration of the whole term of service. How much is due to the workman, allowing simple interest at 6 per cent. per annum?

ANALYSIS. Here the last month's wages, $20, is the last term; the number of months, 18, is the number of

terms; and the interest on 1 month's wages, $.10, is the common difference; and since the first month's wages has been on interest 17 months, the progression is a descending series. Then, by 706 we find the first term, which is the amount of the first month's wages for 17 months; and by 709 we find the sum of the series, which is the sum of all the wages and interest.

2. A father deposits annually for the benefit of his son, commencing with his tenth birthday, such a sum that on his 21st birthday the first deposit at simple interest amounts to $210, and the sum due his son to $1860. How much is the deposit, and at

what rate per cent. is it deposited?

terms; and $1860, the amount of all the deposits and interests, is the sum of the series. By 709 we find the last term to be $100, which is the annual deposit; and by 707 we find the common difference to be $10, which is the annual rate %.

3. What is the amount of an annuity of $150 for 5 years, payable quarterly, at 1 per cent. per quarter? Ans. $3819.75.

4. In what time will an annual pension of $500 amount to $3450, at 6 per cent. simple interest?

Ans. 6 years. 5. Find the rate per cent at which an annuity of $6000 will amount to $59760 in 8 years, at simple interest.

Ans. 7 per cent.

ANNUITIES AT COMPOUND INTEREST.

727. An Annuity at compound interest constitutes a geometrical progression whose first term is the annuity itself; the common multiplier is one plus the rate per cent. for one interval expressed decimally; the number of terms is the number of intervals for which the annuity is taken; and the last term is the first term multiplied by one plus the rate per cent. for one interval raised to a power one less than the number of terms.