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first term of the old is not found in the new. Now, if the old series except the last term be subtracted from the new, the remainder will be the difference of the extremes in the old series, the other terms in the two series canceling each other; the remainder will also be 3 times the sum of all the terms except the last in the old series; for once a series from 4 times a series must leave 3 times the series; ..of this remainder plus the last term must be the sum of all the terms in the old series; but 3 is the ratio less 1.

A similar explanation is always applicable. Hence,

363. PROB. 3. The extremes and ratio being given, to find the sum of the series,

RULE. Divide the difference of the extremes by the ratio less 1, and to the quotient add the greater extreme.

Ex. 1. The extremes are 2 and 20000, and the ratio 10; what is the sum of the series?

20000 — 2 — 19998; 10—1=9; 199989: 222220000 = 22222, Ans.

2222; and

2. The extremes are 7 and 45927, and the ratio 3; what is the sum of the series?

Ans. 68887.

3. The extremes of a series are 5 and 5120, and the ratio 4; what is the sum?

364. PROB. 4.

Ans. 6825.

The first term, ratio and number of terms

being given, to find the sum of the series,

RULE.

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Find the last term by Prob. 1, and the sum of the series by Prob. 3.

Ex. 1. The first term is 5, the ratio 3 and the number of terms 9; what is the sum of the series?

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32805, last term; (32805

5) ÷ 2 =

38 X 5 16400, difference of extremes divided by the ratio less one; 16400 + 32805 49205, sum of the series.

2. The first term is 7, ratio 6 and the number of terms 13; what is the sum of the series? Ans. 18284971621.

3. A lady being married on the first day of January, her father gave her $1, promising to give her $10 on the first of February, and so on in geometrical series on the first of the remaining months of the year; to what sum did her dowry amount? Ans. $111111111111.

4. Had the ratio been 5 instead of 10 in the above example, what would have been the lady's dowry? Ans. $61035156.

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365. AN ANNUITY is, properly, a sum of money payable annually; but the term is also applied to sums payable monthly, quarterly, semi-annually, biennially, or at any regular intervals. Rents, salaries, pensions, etc., are annuities.

366. An annuity payable at a definite number of times is called a certain annuity; if payable periodically for an indefinite time, e. g. during the life of an individual, it is a contingent or life annuity; if payable at regular intervals forever, e. g. the interest of a school fund, it is a perpetual annuity or a perpetuity.

367. An annuity already commenced, or to commence immediately, is said to be in possession; if not to commence until a definite time has elapsed, or until the occurrence of a specified event, it is in reversion; if not paid when it becomes due, it is in arrears.

368. The amount of an annuity in arrears is the sum of all the instalments which are due and unpaid, together with all the interest which has arisen on such instalments.

369. The right to some species of annuities may be bought; in such cases, the purchase money is the present worth of the annuity.

370. The amount of annuities in arrears, at simple and at compound interest, may be found in various ways.

371. PROB. 1.—To find the amount of an annuity at simple interest,

Ex. 1. What is the amount of an annuity of $100 per year in arrears for 4 years, on simple interest at 6 per cent. per annum?

$100 + $106 + $112 + $118

1ST METHOD. The 4th instalment, becoming due to-day, is worth just $100; the 3d instalment, having been due 1 year, amounts to $106; so the 2d and 1st instalments, having been due 2 and 3 years, respectively, amount to $112 and $118; .. $436, the sum sought; but these numbers constitute an arithmetical series, of which the first term is the annuity, the common difference is the interest of the annuity at the given per cent. for the time between two successive payments, and the number of terms is the number of payments; .. we find the amount of the annuity by the rule in Art. 351.

2D METHOD.-As the several instalments are on interest for 1, 2 and 3 years, it is plain that the entire interest is equal to the interest of $100 for 1 year multiplied by (1 + 2 + 3) ; i. e. the entire interest = $6 × 6 $36, and this added to the sum of the 4 instalments, viz., $400, gives $136, as in the 1st method. Hence,

=

RULE. Find the sum of the natural series of numbers, 1, 2, 3, etc., up to the number of instalments, less one, by Art. 351; multiply the interest of one instalment for one interval of time, by this sum, and the product will be the entire interest; add the entire interest to the sum of all the instalments and the whole sum will be the amount required.

Ex. 2. If an annual pension of $500 be in arrears for 6 years, what will it amount to at 6 per cent. simple interest?

Ans. $3450. 3. What is the amount of a salary of $225 quarterly, in arrears for 4 years, at 6 per cent. per annum, simple interest? Ans. $4005.

4. What is the amount of an annual salary of $6000, in arrears for 8 years, at 7 per cent. simple interest.

Ans. $59760.

5. If a semi-annual rent of $350 be in arrears for 3 years and 6 months, what will it amount to at 8 per cent. simple interest?

6. The interest on a certain sum is $600 per annum; if this interest remains unpaid for 3 years, what, in justice, would be due the creditor, money being worth 10 per cent.?

7. What, money being worth 6 per cent.?

372. PROB. 2.-To find the amount of an annuity in arrears at compound interest,

Ex. 1. What is the amount of $1 annuity per annum, in arrears for 4 years, at 6 per cent. compound interest?

The 4th instalment, becoming due to-day, is worth just $1; the 3d, having been due 1 year, is worth $1.06; so the 2d and 1st instalments, having been due for 2 and 3 years respectively, amount, at compound interest, to $1.1236 and $1.191016; .. $1.+ $1.06 + $1.1236 + $1.191016 $4.374616, the sum sought; but these numbers constitute a geometrical series, of which the first term is the annuity, the ratio is the amount of $1 at the given rate for the time between two successive payments, and the number of terms is the number of payments; .. we find the amount of the annuity by the rule in Art. 364.

Ex. 2. If an annual pension of $500 be in arrears for 6 years, what will it amount to at 6 per cent. compound interest? Ans. $3487.6592688.

REMARK.-In the several Problems in Annuities $1 may be considered the annuity, and having proceeded with $1 according to the rule, the product of the result multiplied by the true annuity will give the true result. Hence the utility of the following

TABLE,

Showing the amount of the annuity of $1, £1, etc., at 4, 5, 6 and 7 per cent. compound interest, for any number of years not exceeding 20.

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11

12

13

14

15

16

17

17.888451

13.486351 14.206787 14.971643
15.025805 15.917127 16.869941
16.626838 17.712983 18.882138 20.140643
18.291911 19.598632 21.015066 22.550488
20.023588 21.578564 23.275970 25.129022
21.824531 23.657492 25.672528
23.697512 25.840366 28.212880

27.888054

15.783599

30.840217

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Ex. 3. What is the amount of an annual pension of $900 in arrears for 18 years, at 7 per cent. compound interest?

Ans. $27756.1953.

4. What is the amount of an annual salary of $1000 which has been in arrears 20 years, at 5 per cent. compound interest? Ans. $33065.954.

5. What is the amount of an annual rent of $150, in arrears for 12 years, at 6 per cent. compound interest?

Ans. $2530.49115. 6. What is the amount of an annuity of $300 per annum, in arrears for 15 years, at 4 per cent. compound interest?

Ans. $6007.0764.

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