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If the last term had been given and the first required, the process would evidently have been by division, since every less term is the result of a division of the term next larger; by ratio. Hence the

RULE. - Raise the ratio to a power whose index is equal to the number of terms less one; then multiply this power by the first term to find the last, or divide it by the last term to find the first.

NOTE.

This rule may be applied in computing compound interest, the principal being the first term, the amount of one dollar for one year the ratio, the time, in years, one less than the number of terms, and the amount the last term.

EXAMPLES FOR PRACTICE.

1. The first term of a series is 1458, the number of terms 7, and the ratio; what is the last term? Ans. 2.

OPERATION.

726; 785 × 1458 1458 2, the last term.

Ratio (3)

2. If the first term of a series is 4, the ratio ber of terms 7, what is the last term?

=

5, and the numAns. 62500.

3. If the first term of a series is 28672, the ratio, and the number of terms 7, what is the last term? Ans. 7.

4. The first term of a series is 5, the ratio 4, and the number of terms is 8; required the last term. Ans. 81920.

5. If the first term of a series is 10, the ratio 20, and the number of terms 5, what is the last term? Ans. 1600000.

6. If the first term of a series is 30, the ratio 1.06, and the number of terms 6, what is the last term?

Ans. 40.146767328. 7. What is the amount of $1728 for 5 years, at 6 per cent., compound interest? Ans. $2312.453798+.

8. What is the amount of $328.90 for 4 years, at 5 per cent., compound interest? Ans. $399.78+.

9. A gentleman purchased a lot of land containing 15 acres, agreeing to pay for the whole what the last acre would come

312. What is the rule for finding a required extreme, when one of the extremes, the ratio, and number of terms are given? To what may this rule be applied?

to, reckoning 5 cents for the first acre, 15 cents for the second, and so on, in a threefold ratio. What did the lot cost him? Ans. $239148.45.

313. To find the SUM OF ALL THE TERMS, the first term, the ratio, and the number of terms being given.

Let it be required to find the sum of the

ILLUSTRATION. following series:

6, 18, 54, 2, 6, 18, 54,

2, 0, 0, 0,

2, 6, 18, 54.

If we multiply each term of this series by the ratio 3, the products will be 6, 18, 54, 162, forming a second series, whose sum is three times the sum of the first series; and the difference between these two series is twice the sum of the first series. Thus,

=

162, the second series.
the first series.

2160, difference of the two series.

Now, since this difference is twice the sum of the first series, one half this difference will be the sum of the first series; thus 160 ÷ 2

€80.

162

It will be observed, that if we had multiplied 54, the last term of the first series, by the ratio 3, and from the product, 162, subtracted 2, the first term, we should have obtained 160; and this being divided by the ratio 3, less 1, would have given 80 for the sum of the first series, as before. Hence the

RULE. - Find the last term as in Art. 312. Multiply by the ratio, and from the product subtract the first term. Then divide this remainder by the ratio less 1, and the quotient will be the sum of the series.

NOTE 1. If the ratio is less than 1, the product of the last term, multiplied by the ratio, must be subtracted from the first term; and, to obtain the divisor, the ratio must be subtracted from unity, or 1.

NOTE 2. When a descending series is continued to infinity, it becomes what is called an INFINITE SERIES, whose last term must be regarded as 0, and its ratio as a fraction. To find the sum of an infinite series,

313. The rule for finding the sum of all the terms, the first term, ratio, and number of terms being given? If the ratio is less than a unit, what must be done with the product of the last term multiplied by the ratio? How is the divisor obtained when the ratio is less than 1?

Divide the first term by 1, decreased by the fraction denoting the ratio, and the quotient will be the sum required.

EXAMPLES FOR PRACTICE.

1. If the first term of a series is 12, the ratio 3, and the num ber of terms 8, what is the sum of the series. Ans. 39360.

OPERATION.

Ratio 37 X 1226244, the last term; 26244 × 3 78732; 78732-12 = 78720; 78720 ÷ (3 — 1) — 39360, the sum of the series.

=

2. The first term of a series is 5, the ratio, and the number of terms 6; required the sum of the series. Ans. 131.

OPERATION.

(3)5

Ratio (4) X 58, the last term; 8 X 3=428; 5 -328-3325; 3325 ÷ (1—3) — 3325 — 13166, the sum of - = the series.

=

3. If the first term of a series is 8, the ratio 4, and the number of terms 7, required the sum of the series. Ans. 43688.

4. If the first term is 10, the ratio, and the number of terms 5, what is the sum of the series? Ans. 30,

5. If the first term is 18, the ratio 1.06, and the number of terms 4, what is the sum of the series? Ans. 78.743+.

6. When the first term is $144, the ratio $ 1.05, and the number of terms 5, what is the sum of the series?

Ans. $795.6909.

7. D. Baldwin agreed to labor for E. Thayer for 6 months. For the first month he was to receive $ 3, and each succeeding month's wages were to be increased by of his wages for the month next preceding; required the sum he received for his 6 months' labor. Ans. $9177.

8. If the first term of a series is 2, the ratio 6, and the number of terms 4, what is the sum of the series? Ans. 518.

9. A lady, wishing to purchase 10 yards of silk for a new dress, thought $1.00 per yard too high a price; she, however, agreed to give 1 cent for the first yard, 4 for the second, 16 for the third, and so on, in a fourfold ratio; what was the cost of the dress? Ans. $3495.25.

ANNUITIES AT COMPOUND INTEREST.

314. An Annuity is at Compound Interest when compound inter. est is reckoned on the annuity in arrears.

The several payments form a geometrical series, of which the annuity is the first term, the amount of $1.00 for one year the ratio, the years the number of terms, and the amount of the annuity the sum of the series.

315. To find the amount of an annuity at compound interest.

RULE 1..

Multiply the amount of $1.00, for the given time and

RULE 2. rate found in the table, by the annuity, and the product will be the required amount.

Years.

1234567a

- Find the sum of the series, as in Art. 313. Or,

TABLE,

SHOWING THE AMOUNT OF $1 ANNUITY AT COMPOUND INTEREST, FROM

1 YEAR TO 40.

11 12

13

14

15

16

17

19

20

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5 per cent.

21

35.719252

22

38.505214

23

41.430475

24

44.501999

25

47.727099

26

51.113454

27

54.669126

28

58.402583

29

62.322712

30

66.438847

31

70.760790

32

75.298829

33

80.063771

34

85.066959

35

90.220307

36

95.836323

37 101.628139

38 107.709546

39 114.095023
40 120.799774

first rule for finding the amount of an the table show?

Years.

6 per cent.

39.992727 43.392290 46.995828

50.815577

54.864512

59.156383

63.705766

68.528112

73.639798

79.058186

84.801677

90.889778

97.343165

104.183755

111.434780

119.120867

127.268119

135.904206

145.058458 154.761966

314. When is an annuity said to be at compound interest? What do the amounts of the several payments form? What is the first term of the series? The ratio? The number of terms? The sum of the series?

annuity? The second?

315. The

What does

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EXAMPLES FOR PRACTICE.

1. What will an annuity of $378 amount to in 5 years, at 6 per cent. compound interest? Ans. $2130.821+.

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2. What will an annuity of $1728 amount to in 4 years, at 5 per cent. compound interest? Ans. $7447.896+.

3. What will an annuity of $87 amount to in 7 years, at 6 per cent. compound interest? Ans. $730.263+.

4. What will an annuity of $500 amount to in 6 years, at 6 per cent. compound interest? Ans. $3487.659+.

5. What will an annuity of $96 amount to in 10 years, at 6 per cent. compound interest? Ans. $1265.356+.

6. What will an annuity of $1000 amount to in 3 years, at 6 per cent. compound interest? Ans. $3183.60.

7. July 4, 1842, H. Piper deposited in an annuity office, for his daughter, the sum of $56, and continued his deposits each year, making the last July 4, 1848. Required the sum in the office July 4, 1848, allowing 6 per cent. compound interest. Ans. $470.054+.

8. C. Greenleaf has two sons, Samuel and William. On Samuel's birthday, when he was 15 years old, he deposited for him, in an annuity office, which paid 5 per cent. compound interest, the sum of $25, and this he continued yearly, making, however, the last deposit on his becoming 21 years of age. On William's becoming 12 years old, he deposited for him, in an office which paid 6 per cent. compound interest, the sum of $20, and continued this yearly, making the last deposit on his becoming 21 years of age. Which will receive the larger sum, when 21 years of age? Ans. $ 60.065+ William receives more than Samuel.

9. I gave my daughter Lydia $10 on her becoming 8 years old, and the same sum on her birthday each year, giving the last on her becoming 21 years old. This sum was deposited in the savings bank, which pays 5 per cent. annually. Required the amount in the bank for her when she is 21 years of age.

Ans. $195.986+.

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