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series is increasing; or from the first term subtract the same if the series is decreasing.

Exercises.

1. If the ages of 5 persons are in arithmetical progression, the youngest being 15 years old, and the common difference is 2, what is the age of the oldest? Ans. 23 years.

-2. A merchant bought 34 yards of cloth, and agreed to give 12 cents for the first yard, 123 cents for the second, 12 cents for the third, and so on; what did the thirty-third yard cost him? Ans. 22 cents.

3. I have 40 books; the first is worth $1.80, the second $1.77, the third $1.74, and so on, each being worth 3 cents less than the preceding; what is the value of the last book? Ans. $.63.

446.

CASE II.

To find the common difference and the number of terms.

Let 3, 5, 7, 9, 11 be an arithmetical series.

32 x 4, and sub

Then, by the preceding case, the last term 11 tracting the first term 3, we have 2 × 4, or the product of the common difference by the number of terms less one.

Hence, to find the common difference,

Divide the difference of the extremes by the number of terms less one.

Also, to find the number of terms,

Divide the difference of the extremes by the common differ ence, and add one to the quotient.

Exercises.

1. The extremes of an arithmetical series are 5 and 271⁄2, and the number of terms 11; what is the common difference?

Ans. 21

2. The extremes of an arithmetical series are 5 and 27, and the common difference 21; what is the number of terms?

How can we find the common difference? The number of terms?

3. A person traveling went the first day 3 miles, and increased his speed every day by 5 miles, till at last he went 58 miles in one day; how many days did he travel? Ans. 12 days.

CASE III.

447. To find the sum of all the terms of an arith metical series.

6,

4,

12, be an arithmetical series,
2, be the same inverted.

Let 2, 4, 6, 8, 10, and 12, 10, 8, Then 14+14 +14 +14 +14 + 14 = 84, the sum of both series. But 84 is equal to 14, the sum of the extremes, multiplied by 6, the number of terms; and half of 84, or of 14 X 6, is the sum of one of the series. Hence, to find the sum of all the terms,

Multiply the sum of the extremes by the number of terms, and take half the product.

Exercises.

1. The clocks of Venice strike from 1 to 24; how many strokes do one of these clocks make in one day? Ans. 300.

2. If a person on a journey travel the first day 30 miles, and each succeeding day a quarter of a mile less than he did the day before, how far will he travel in 30 days? Ans. 7914 miles.

3. If 100 eggs be laid one yard distant from one another in a straight line, and a basket be placed one yard from the first one, what distance must a person travel to gather them singly and place them in the basket? Ans. 5 m. 1300 yd.

GEOMETRICAL SERIES.

448. A Geometrical Series, or Progression, is a series in which the terms vary by a common multiplier.

3, 9, 27, 81, 243,

Thus,

is an increasing geometrical series, in which 3 is the common multiplier.

How do we find the sum of all the terms? What is a Geometrical Series?

The RATE, or RATIO, of a geometrical series is the common multiplier.

CASE I.

449. To find any term in a geometrical series.

Let 4 the first term, 2 = the rate, and 5 = the number of terms. Then,

2d term 4 X 2; 3d term = 4 × 22; 4th term = 4 × 23; 5th term = 4 X 24.

That is, when the first term, the rate, and the number of terms are given, to find any term of the series,

Multiply the first term by the rate raised to a power whose exponent is equal to the number of terms which precede the required term.

Exercises.

Ans. 4.

1. Find the 8th term of a geometrical series whose first term is 6 and rate 2. Ans. 768. 2. Find the 6th term of a geometrical series whose first term is 4096 and rate 3. A gentleman dying left 11 sons, to whom he bequeathed his property, as follows: to the youngest he gave $1024; to the next, as much and a half; to the next, 14 of the preceding son's share, and so on. What was the eldest son's portion?

CASE II.

Ans. $59049.

450. To find the rate of a series.

32,

Let 4, 8, 16, 64, be a geometrical series. Then, by the preceding case, the 5th term 64 = 4 × 2a, and divid }ng by the first term, we have 2*, or the fourth power of the rate. Hence, to find the rate of a geometrical series,

Divide the last term by the first, and extract that root of the quotient whose index is denoted by the number of terms less one.

What is the Rate or Ratio of a geometrical series? How do you

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Exercises.

1. Find the rate of a geometrical series whose first term is 6, last term 768, and number of terms 8.

Ans. 2.

2. Find the rate of a geometrical series whose first term is 4096, last term 4, and number of terms 6.

Ans.

3. A man paid a debt by making 12 payments in geometrical progression, the first payment being $3, and the last $6144; what was the rate?

CASE III.

Ans. 2.

451. To find the sum of all the terms of a geometrical series.

Let 4, 12, 36, 108, be a geometrical series; and multiplying it by the rate 3, we have

a second series

the first series,

twice the sum of the first=

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4, 0, 0,

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Subtracting, we have

0, 324, or 324 — 4; that is, the sum of the first: 324-4, which is the difference between the first term and the product of the last term by the rate, divided by the rate less one. Hence, to find the sum of all the terms,

Multiply the last term by the rate, subtract the first term from the product, and divide the result by the rate less one.

Exercises.

1. Find the sum of a geometrical series whose extremes are 2 and 128, and rate 4. Ans. 170.

2. If the descendants of the 101 persons who landed at Plymouth, in the year 1620, had increased so as to double their number in every 20 years, how great would have been the aggregate to the year 1860? Ans. 413595

3. A jockey offered to sell some fine horses to a young man not well versed in numbers, and receive in payment $1 for the first, $3 for the second, $9 for the third, and so on. The young man, thinking it a great bargain, agreed accordingly; what did the horses cost him, provided there were 12 of them?

How do we find the sum of all the terms?

Ans. $265720

ANNUITIES.

452. An Annuity is a fixed sum of money payable at the end of equal periods of time.

An annuity is said to be forborne, or in arrears, when payments are not paid when due.

453. The AMOUNT, or FINAL VALUE, of an annuity is the sum of the amounts of all its payments at interest from the time each becomes due.

454. The PRESENT VALUE of an annuity is such a sum as, put at interest, will, for the given time and rate, exactly amount to the annuity.

Pensions, rents, reversions, life insurance, etc., involve the principle of annuities.

CASE I.

455. To find the amount of an annuity at simple interest.

1. Required the amount of an annuity of $100, forborne five years, at 6 % simple interest.

At the end of the 5th year there will be due: the 5th year's payment, or $100; the 4th year's payment, $100, plus 1 year's interest, or $106; the 3d year's payment, $100, plus 2 years' interest, or $112; the 2d year's payment, $100, plus 3 years' interest, or $118; and the 1st year's payment, $100, plus 4 years' interest, or $124.

Hence, the sums due are $100 + $106+$112+$118+ $124, or $560.

But the sums due at the end of the 5th year form an arithmetical series, of which the annuity, or $100, is the first term, its interest for 1 year, or $6, is the common difference, and the number of years, or 5, is the number of terms. Hence,

Find the amount of the first payment for the last term of an arithmetical series, and then the sum of the series for the amount of the annuity.

arrears?

What is an Annuity? When is an annuity said to be forborne, or in What is the Amount of an annuity? The Present Value of an annuity? How do we find the amount of an annuity at simple interest?

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