GEOMETRICAL PROGRESSION.

491. A series of numbers which increase or decrease by a constant multiplier or ratio is called a Geometrical Progression.

Thus 5, 10, 20, 40, 80, etc., is a geometrical progression, of which the multiplier or ratio is 2.

WRITTEN EXERCISES.

492. 1. The first term of

a geometrical series is 3 and

the multiplier or ratio is 2. What is the 5th term?

24=16

3 x 16 = 48

EXPLANATION. Since the multiplier is 2, the second term will be 3 x 2, the third 3 × 2 × 2 or 3 x 22, the fourth 3 x 22 × 2 or 3 × 23, and the fifth 3×23×2 or 3 x 24, that is, the fifth term is equal to the first term multiplied by the ratio raised to the fourth power.

RULE.

Any term of a geometrical progression is equal to the first term, multiplied by the ratio raised to a power one less than the number of the term.

2. The first term of a geometrical progression is 10, and the ratio 3. What is the 6th term?

3. The first term of a geometrical progression is 10, the ratio 4, and the number of terms 6. What is the 6th term?

4. If a farmer should hire a man for 10 days, giving him 5 cents for the first day, 3 times that sum for the second day, and so on, what would be his wages for the last day?

5. If the first term is $100 and the ratio 1.06, what is the 6th term? Or, what is the amount of $100 at compound interest for 5 years at 6%?

6. What is the amount of $520 for 6 years, at 5% compound interest?

7. What is the sum of a geometrical series, of which the first term is 5, the ratio 3, and the number of terms 5?

EXPLANATION.- Since in this series the first term is 5, the ratio 3, and the number of terms 5, their sum may be obtained by the fol

lowing process, which illustrates the formation of the rule:

Series 5+ 15 + 45 + 135 + 405

3 times Series 15+45 + 135 + 405 + 1215

2 times Series = 1215 5

RULE.

The sum of a geometrical series is equal to the difference between the first term, and the product of the last term by the ratio, divided by the difference between the ratio and 1.

Or, since the last term is equal to the first term multiplied by the ratio raised to a power one less than the number of terms,

The sum of a geometrical series is found by dividing the difference between the first term and the first term multiplied by the ratio raised to the power equal to the number of the terms by the difference between the ratio and 1.

8. The extremes of a geometrical progression are 4 and 1024, and the ratio is 4. What is the sum of the series?

9. The extremes are and 3 and the ratio is 21. What is the sum of the series?

10. What is the sum of the series in which the first term is 2, the last term 0, and the ratio ; or what is the sum of the infinite series 2, 1,,,, 16, 32, etc.?

11. The extremes of a geometrical progression are and 15625, and the ratio is 14. What is the sum of the series? 12. If a child should receive 1 cent at birth, 2 cents on the second birthday, 4 cents on the third, etc., how much would he be worth when 21 years of age?

PROBLEMS IN COMPOUND INTEREST.

WRITTEN EXERCISES.

493. To find the principal, when the compound interest, the time, and the rate are given.

1. What principal at 6% compound interest will produce $2372.544 interest in 10 years? $1.790848 $1

= $.790848.

$2372.544.790848 = $3000.

EXPLANATION.

-By geomet

rical progression, or by the compound interest table on page 272, the amount of $1 at compound interest for the given time at the given rate is found to be $1.790848. That sum less $1 gives $.790848, the compound interest of $1 for the given time at the given rate. Then, $2372.544 ÷ .790848 = $3000, the principal.

2. What principal at 6% compound interest will produce $3150 interest in 8 years?

3. What principal at 5% compound interest will produce $2896 interest in 12 years ?

4. What principal at 7% compound interest will produce $3600 interest in 15 years? At 4% in 20 years?

494. To find the rate, when the principal, compound interest, and time are given.

1. At what rate per cent will $500 yield $203.55 compound interest in 7 years?

$203.55+500-$.4071.

EXPLANATION. - Since $203.55 is the compound interest of $500 for 7 years, of that sum will be the compound interest of $1 for the same time. By referring to the compound interest table, opposite 7 years, we find the amount $1.4071, or the interest $.4071, in the 5% column. Therefore, the rate is 5%.

2. At what rate per cent will $1000 yield $503.63 compound interest in 7 years?

3. At what rate per cent will $1200 yield $721.2384 compound interest in 12 years?

4. What is the rate per cent when $1800 yields $901.314 compound interest in 6 years?

5. What is the rate per cent when $2000 yields $4344.338 compound interest in 15 years?

495. To find the time when the principal, the compound interest, and rate are given.

1. In what time will $600 amount to $1200 at 7% compound interest?

137701 yr. = 2 mo. 26 da.

... The time =

10 yr. 2 mo. 26 da.

EXPLANATION. - Since $600 amounts to $1200 at 7% in a certain time, $1 in the same time and at the same rate, will amount to of $ 1200, or $2.

By the compound interest table, $1 at 7% will in 10 yr. amount to $1.967151, and in 11 yr. to $2.104852, consequently the time must be between 10 and 11 yr. The interest of $1.967151 for a year at 7% is $.137701, and the difference between $2 and $1.967151 is $.032849. Since the interest of $1.967151 for a year is $.137701, to earn $.032849 will require 32849 of a year, or 2 mo. 26 da. Therefore the time is 10 yr. 2 mo. 26 da.

2. In what time will $400 amount to $1000, at 6% compound interest?

3. In what time will $750 amount to $1500, at 5% compound interest? Or, in how long a time will any sum double itself at 5% ?

4. In what time will $960 amount to $2000, at 7% compound interest?

5. In what time will $1300 amount to $2500, at 6% compound interest?

6. In what time will $3200 amount to $4800, at 4% compound interest?

ANNUITIES.

496. A definite sum of money payable at the end of equal periods of time is an Annuity.

Properly speaking, an annuity is a sum payable annually, but sums payable at intervals of quarter years, half-years, or other periods, are also called annuities.

497. An annuity which continues forever is called a Perpetual Annuity or Perpetuity.

498. An annuity which commences at a definite time, and continues for a definite time, is called a Certain Annuity.

499. An annuity whose commencement or continuance, or both, depend upon some contingent event, as the death of some person, is called a Contingent Annuity.

500. An annuity upon which the payments were not made when they were due is called an Annuity in Arrears or Forborne.

501. The Amount or Final Value of an annuity is the sum of all the payments, increased by the interest of each payment, from the time it becomes due until the annuity ceases.

502. A sum of money, which, upon being put at interest for the given time at the given rate, will be equal to the amount of the annuity, is the Present Value of the annuity.

503. Annuities are sometimes computed at Simple Interest and sometimes at Compound Interest.