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EXAMPLES.

1. What is the seventh term of the series 2, 6, 18, 54, &c.? 2. What is the fifteenth term of the series 5, 2, 11, §, 16, &c.? 3. What is the amount of $500 for 7 years, at 6 per cent., compound interest?

NOTE.-1.06 is the ratio, and the amount the eighth term of the series. 4. Naturalists have found that the ratio of increase of some kinds of animalculæ (microscopic animals) is often four in a single day. At that rate, what would be the increase of one animalcula and its descendants in ten days? Ans. 1,048,576.

496. TO FIND THE RATIO, THE FIRST TERM, THE LAST TERM, AND NUMBER OF TERMS BEING GIVEN.

In series (1), if the last term, 3 X 24, be divided by the first term, 3, the quotient will be 24, or the fourth power of the ratio, the fourth root of which will equal the ratio. Hence,

To find the ratio: Divide the last term by the first term, and extract that root of the quotient whose index equals the number of terms less one.

EXAMPLES.

5. The first term of a series is 2, the last term 128, the num ber of terms 3; what is the ratio?

Ans. 8. 6. The first term is 4, the last term, and the number of terms 4; what is the ratio?

Ans.

7. The extremes are 5 and 625, and the number of terms 4; what is the ratio?

497. TO FIND THE SUM OF A SERIES.

Let 3, 9, 27, 81, 243, be a series, of which we wish to find the

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By multiplying each term of the series by the ratio, 3, we have a second series, whose sum is 3 times that of the first series, from which we subtract the first series; the remainder equals twice the sum of the first series, which we find by dividing by 2. Hence,

To find the sum of the series: Subtract the first term from the product of the last term multiplied by the ratio; divide the remainder by the ratio less one.

NOTE. If the series is descending, the last term multiplied by the ratio should be taken from the first term, and the remainder be divided by one less the ratio.

EXAMPLES.

8. What is the sum of the series 3, 12, 48, 192, 768, 3072?

Ans. 4095. 9. The first term is 5, the last term 3125, and the number of Ans. 3905. terms 5; what is the sum of the series? 10. What is the sum of 7 terms of the series 4, 8, 16, 32, &c.? 11. If of the air in a receiver be taken from it by an airpump at the first stroke of the piston, and of the remainder at the second stroke, and so on, what will be the amount taken Ans. from the receiver by 8 strokes?

ANNUITIES.

498. Annuities are periodical payments of fixed sums of money, in consideration of money paid or services rendered.

499. When an annuity is made for a definite number of years, it is called a certain annuity; when it is made forever, a perpetuity; when it depends upon the life of one or more persons, a life annuity; when it does not commence till a given time has elapsed, it is said to be in reversion.

500. When annuities are granted by government, they are called Pensions.

501. The Amount of an annuity is the sum of all the payments, plus their interest, from the time they become due.

502. The Present Worth of an annuity is such a sum of money as, put at interest, will exactly pay the annuity.

503. Annuities are said to be in Arrears, or Foreborne, when they remain unpaid after they become due.

est.

504. Annuities are generally computed at compound inter

ANNUITIES AT SIMPLE INTEREST.

505. ILL. EX. What is the amount of an annuity of $200 a year, at 6 per cent. simple interest, 5 years in arrears ?

The payment due at the end of the fifth year is $200; that which was due at the end of the fourth year amounts, at the end of the fifth year, to $200 plus the interest on the same for 1 year; that which was due at the end of the third year, to $200 plus its interest for 2 years; that due at the end of the second year, to $200 plus its interest for 3 years; that due at the end of the first year, to $200 plus its interest for 4 years. Hence, the sums due at the end of the fifth year would form an arithmetical series, 200, 200+ 12, 200+ 24, 200 +36, 200+ 48, of which the first term is $200, the last term the amount of $200 for the number of years less 1, and the number of terms the number of years. Hence the sum may be found by Art. 490; and, generally,

To find the amount of an annuity at simple interest: Find the sum of an arithmetical series, of which the first term is the last payment, the last term the amount of the first payment, and the number of terms the number of payments.

EXAMPLES.

1. What is the amount of an annuity of $300 for 6 years, at 6 per cent., simple interest? Ans. $2070. 2. What is the amount of an annuity of $600 for 7 years, at 7 per cent., simple interest?

3. A gentleman's salary of $1200 a year, payable quarterly, remained unpaid for 4 years; what was then his due?

ANNUITIES AT COMPOUND INTEREST.

506. İLL. EX. What is the amount of an annuity of $36 for 4 years, at 6 per cent., compound interest?

We will first find the amount of an annuity of $1 for the same time. The last payment, due at the end of 4 years, will be $1. The sum due on the third payment, at the end of the fourth year, will be the amount of $1 for 1 year; that due on the second payment will be the amount of $1 for 2 years; that due on the first payment will be the amount of $1 for 3 years. Hence the four sums due will form the geometrical series,

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1.19101, or

1, 1.06, 1X (1.06)2 1 X (1.06)3,

of which the first term is the last payment, the last term the amount of the first payment, and the number of terms the number of payments. Finding the sum of this series (Art. 297), and multiplying by 36, we obtain the required amount. Hence, the

RULE. To find the amount of an annuity at compound interest: Find the amount of an annuity of $1 for the given time by geometrical progression (Art. 497), and multiply the sum thus abtained by the annuity.

EXAMPLES.

1. What is the amount of an annuity of $1 for 2 years, at 6 per cent.? for 3 years? for 5 years? for 10 years?

2. What is the amount of an annuity of $20 for 8 years, at 5 per cent.? Ans. $190.98.

507. TABLE I.,

Showing the amount of $1 or £1 annuity from 1 year to 20.

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NOTE.-The following examples may be performed by the use of

Table I.

What are the amounts of the following annuities?
3. $100 for 7 years, at 5
per cent.
4. $200 for 10 years, at 6 per cent.
5. £150 for 18 years, at 6 per cent.

6. A gentleman, on his daughter's first birthday, and on each succeeding birthday, deposited $10 in a savings-bank, which yielded 5 per cent. compound interest, and presented her with the amount on her eighteenth birthday. What was the value of the present?

508. To find the present worth of an annuity: Divide the amount of the annuity by the amount of $1 compound interest for the time given. It may also be obtained by the use of the following table:

TABLE II.,

Showing the present value of an annuity of $1 or £1 from

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7. What is the present worth.of an annuity of $200 for 4 cent.? per years, at 5

Ans. $709.19. 8. What must I pay for an annuity of $300 for 10 years, at 6 per cent.?

509. QUESTIONS FOR REVIEW.

Of what does ALLIGATION treat? What is Alligation Medial? Al ligation Alternate? What other name might be used for Alligation? Make an example in Alligation Medial; perform and explain it, and

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