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RULE. Multiply the first term by that power of the ratio whose index is one less than the number of terms, and the product will be the last term; or, divide the last term by the same power of the ratio, and the quotient will be the first term.

2. The first term of a geometrical series is 4, the ratio 5, and the number of terms 6; what is the last term?

3. The last term of a series is 531441, the ratio 3, and the number of terms 12; what is the first term?

4. The last term is 63.123848, the ratio 1.06, and the number of terms 5; what is the first term?

5. What is the amount of $80 at compound interest for 5 years, the rate being 6 per cent. ?

Note.—The ratio is 1.06, the number of terms 5+1= 6, and the amount is the last term.

6. What is the amount of $10000 at 5 per cent. compound interest for 10 years?

7. The first term is 10, the ratio 1, and the number of terms 6; what is the last term?

CASE II. To find the ratio, when the extremes and the

number of terms are given.

1. The first term is 4, the last term number of terms 5; what is the ratio ?

2500, and the

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ANALYSIS. -As we multiply the first term 4 by the 4th power of the ratio to obtain 2500 the last term, if we divide 2500 by 4, the quotient 625 is the 4th power of the ratio, and extracting the 4th root of this quotient we obtain 5 as the required ratio.

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

2. The extremes are 4 and 256, and the number of *terms is 4; what is the ratio?

3. The first term is 2, the last term 1, the number of terms 6; what is the ratio?

4. The first term is 25, the last term 31.561924, the number of terms 5; what is the ratio?

5. $50 at compound interest will amount to $63.123848 in 4 years; what is the rate per cent. ?

Note.— The number of terms is 5.

CASE III. To find the number of terms, when the ratio and

the extremes are given.

1. The ratio is 5, the extremes are 4 and 2500; what is the number of terms ? 2500

= 625 = 54; 4+1= 5, the number of terms. 4 ANALYSIS.—Dividing the last term by the first, we obtain 625, that power of the ratio which is one less than the number of terms. 625=54, and 4+1=5, the number of terms required.

RULE. Divide the last term by the first, raise the ratio to a power equal to this quotient, and add one to the index of that power for the number of terms.

2. The extremes are 3 and 96, and the ratio is 2; what is the number of terms?

3. The extremes are 25 and 31.561924, and the ratio 1.06; what is the number of terms?

4. In what number of years will $100 amount to $157.351936 at 12 per cent. compound interest?

Note.—The number of terms is one more than the required number of years.

5. The extremes are 4 and 1, and the ratio is }; what is the number of terms?

CASE IV. To find the sum of the series, when the ratio and

the extremes are given. 1. The extremes are 2 and 54, and the ratio is 3; what is the sum of the series? (54 X 3) – 2 F= 80, the sum of the series.

ANALYSIS.—Multiply6 + 18 +54 + 162 = 240 ing each term of the 2 + 6 + 18 + 54

= 80 given series by the ratio

3, we obtain a series the - 2 + 0 + 00 + 00 + 162 = 160 sum of which is 3 times

as great as the sum of the given series. Subtracting the latter series term by term from the former, there remains the difference between the first term of the one and the last term of the other, or (54 X 3) - 2, which is evidently equal to twice the sum of the given series, and the sum of the given series is

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RULE. Multiply the last term by the ratio, and divide the difference between this product and the first term by the difference between the ratio and one.

. 2. The extremes are 3 and 648, and the ratio is 6; what is the sum of the series?

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

4. What is the sum of the infinite series 1, 1, 1, 1, ta, etc.? Note.-The last term may be taken as 0.

5. The extremes are $5 and $1280, and the ratio 4; what is the sum of the series?

ANNUITIES. An Annuity is a sum of money which is payable at regular periods of time.

A Certain Annuity is one that continues for a fixed period of time.

A Perpetual Annuity, or Perpetuity, is one that continues forever.

A Contingent Annuity is one that begins or ends, or begins and ends on the occurrence of certain specified future events, as on the death of one or more persons.

An Annuity in Reversion is one that begins at a specified future time, or on the occurrence of a specified future event.

An Annuity in Arrears, or Forborne, is one the payments of which have been allowed to accumulate, instead of being paid when due.

The Amount of an annuity is the sum of all the payments plus the interest on each payment from the time it becomes due until the final payment.

The Present Worth of an annuity is that sum of money which will, in the given number of payments and at the given rate, amount to the final value.

Pensions and ground-rents are of the nature of annuities. Annuities are usually computed at compound interest. TABLE. The amount of an annuity of $1 for any number of

years from 1 to 25.

YR. 2% 3% 4% 15% 16%

1 $1.000000 $1.000000 $1.000000 $1.000000 $1.000000 2 2.020000 2.030000 2.040000 2.050000 2.060000 3 3.060400 3.090900 3.121600 3.152500 3.183600 4 4.121608 4.183627 4.246464 4.310125 4.374616 5 5.204040 5.309136 5.416323 5.525631 5.637093 6 6.308121 6.468410 6.632975 6.801913 6.975319 77.434283 7.662462 7.898294 8.142008 8.393838 8 8.582969 8.892336 9.214226 9.549109 9.897468 9 9.754628 10.159106 10.582795 11.026564 11.491316 10 10.949721 11.463879 12.006107 12.577893 13.180795 11 12.168715 12.807796 13.486351 14.206787 14.971643 12 13.412090 14.192030 15.025805 15.917127 16.869941 13 14.680332 15.617790 16.626838 17.712983 18.882138 14 15.973938 17.086324 18.291911 19.598632 21.015066 15 17.293417/18.598914 20.023588 21.578564 23.275970 16 18.639285 20.156881 21.824531 23.657492 25.672528 17 20.012071 21.761588 23.697512 25.840366 28.212880 18 21.412312 23.414435 25.645413 28.132385 30.905653 19 22.840559 25.11686827.67122930.539004 33.759992 20 24.297370 26.870374 29.778079 33.06595436.785591 21 25.783317 28.676486 31.969202 35.719252 39.992727 22 27.298984 30.536780 34.247970 38.505214 43.392290 23 28.844963 32.452884 36.617889 41.430475 46.995828 24 30.421862 34.426470 39.082604 44.501999 50.815577 25 32.030300136.459264 41.645908 47.727099 54.864512

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