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geometrical progression, when the first term, the number of terms, and the ratio, are given; this has been done by case V., under geometrical progression. The rule may be stated as follows:

RULE.

From the amount of $1, for 1 year, raised to a power whose exponent is equal to the number of years, subtract $1, divide the remainder by the interest of $1, for 1 year, then multiply the quotient by the annuity.

NOTE. The different powers of the amount of $1, for one year, may be taken from the table under Art. 69.

Examples.

1. What is the amount of an annuity of $200, which has been forborne 14 years, at 6 per cent., compound interest?

From table under Art. 69, we find the 14th power of the amount of $1, for one year, at 6 per cent., to be $2.260904; subtracting $1, and dividing the remainder by $0.06, the interest of $1 for one year, we get $21.01506, which, multiplied by $200, the annuity gives $4203.012, for the amount required.

2. Suppose a person, who has a salary of $700 a year, payable quarterly, to allow it to remain unpaid for 4 years. How much would be due him, allowing quarterly compound interest at 12 per cent. per annum? Ans. $3527.454.

3. What is due on a pension of $150 a year, payable halfyearly, but forborne 2 years, allowing half-yearly compound interest, at 6 per cent. per annum? Ans. $313.772.

4. What is due on a pension of $350 a year, payable quarterly, but forborne 2 years, allowing quarterly compound interest, at 12 per cent.?

Questions under this rule may be easily wrought by the following table.

This table shows the amount of an annuity forborne for any number of years, not exceeding 30, at 3, 4, 5, and 6 per cent, compound interest.

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4

5

6

7

4.310125 4.374616

5.525631 5.637093

4.183627 4.246464
5.309136 5.416323
6.468410 6.632975 6.801913 6.975319
7.662462 7.898294 8.142008 8.393838
8 8.892336 9.514226 9.249109 9.897468
9 10.159106 10.582795 11.026564 11.491316
10 11.463879 12.006107 12.577893 13.180795
11 12.807796 13.486351 14.206787 14.971643
12 14.192030 15.025805 15.917127 16.869941
13 15.617790 16.626838 17.712983 19.882138
14 17.086324 18.291911 19.598632 21.015066
15 18.598914 20.023588 21.578564 23.275970
16 20.156881 21.824531 23.657492 25.672528
17 21.761588 23.697512 25.840366 28.212880
18 23.414435 25.645431 28.132385 30.905653
19 25.116868 27.671229 30.539004 33.759992
20 26.870374 29.778079 33.065954 36.785591
21 28.676486 31.969202 35.719252 39.992727
22 30.536780 34.247970 38.505214 43.392290
23 32.452884 36.617859 41.430475 46.995828
24 34.426470 39.082604 44.501999 50.815577
25 36.459264 41.645908 47.727099 54.864512
26.38.553042 44.311745 51.113454 59.156383
27 40.709634 47.084214 54.669126 63.705766
28 42.930923 49.967583 58.402583 68.528112
29 15.218850 52.966286 162.322712 73.639798
47.575416 56.084938 66.438847 79.058186

30

5. What is due on a pension of $1000, which has been forborne 27 years, at 3 per cent., compound interest?

From the above table we find the amount of an annuity of $1, for 27 years, at 3 per cent., to be $40.709634, which, multiplied by $1000, gives $40709.634, for the amount due.

6. What is the amount of an annuity of $50, which has been forborne 30 years, at 6 per cent., compound interest?

7. What is the amount of a pension of $300, which has been forborne 19 years, at 5 per cent., compound interest? 8. What is the amount of a pension of $900, which has been forborne 17 years, at 4 per cent., compound interest?

9. What is the amount of an annuity of $75, which has been forborne 13 years, at 5 per cent., compound interest?

CASE II.

To find the present worth of an annuity which is to terminate in a given number of years.

The present worth of an annuity, is obviously such a sum of money as will, at compound interest, produce an amount equal to the amount of the annuity. Therefore, if we find the amount of the annuity by Case I., we may consider it as the amount of a certain principal, which principal is the same as the present worth. We have already been taught how to find the present worth, by rule under compound discount. Hence, we have this

RULE.

First, find the amount of the annuity, as if it were in arrears for the whole time, by the aid of the table under Case I., of ANNUITIES.

Then, find the present worth of this amount for the given time and rate per cent., by the use of the table under coмPOUND DIS

COUNT.

Examples.

1. What is the present worth of an annuity of $500, to continue 10 years, interest being 6 per cent. ?

By the table under Case I., of annuities, we find the amount of an annuity of $1, for 10 years, at 6 per cent., to be $13. 180795, this, multiplied by 500, gives $6590.3975, for the amount of the annuity.

Now, by the table under compound discount, we find the present worth of $1, for 10 years, at 6 per cent,, to be $0.558395, which, multiplied by 6590.3975, gives $3680.045, for the present worth required.

2. What is the present worth of an annuity of $100, to continue 20 years, at 5 per cent. interest?

Ans. $1246.222.

The work under this Rule may be very much simplified by

the use of the following table.

The following table gives the present worth of an annuity of $1, or £1, for any number of years, not exceeding 30, at 3, 4, 5, and 6 per cent.

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9.294984

9.712249

14 11.296070 10.563122 9.898641
15 11.936935 11.118387 10.379658
16 12.561102 11.652296 10.837770 10.105895
17 13.166118 12.165669 11.274066 10.477260
18 13.753513 12.659197 11.689587 10.827603
19 14.323799 13.133839 12.085321 11.158116
20 14.877475 13.590326 12.462216 11.469921
21
15.415024 14.029160 12.821153 11.764077
22 15.936917 14.451115 13.163003 12.041582
23 16.443608 14.856842 13.488574 12.303379
24 16.935542 15.246963 13.798642 12.550358
25 17.413418 15.622080 14.093945 12.783356
26 17.876842 15.982769 14.375185 13.003166
27 18.327031 16.329586 14.643034 13.210534
28 18.764108 16.663063 14.898127 13.406164
29 19.188455 16.983715 15.141074 13.590721
30 19.600441 17.292033 15.372451 13.764831

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