Εικόνες σελίδας
PDF
Ηλεκτρ. έκδοση

SECTION II.

GEOMETRICAL PROGRESSION.

310. A Geometrical Progression is a series of numbers varying by a fixed multiplier or factor. Thus, 3, 9, 27, or 27, 9, 3, is a geometrical series. The former, increasing from left to right, is called an ascending series, and the latter a descending series.

311. The rate or ratio of the series is the common multiplier.

312. An Infinite Series is a decreasing series, in which the number of terms is unlimited.

313. The following are the quantities considered in a geometrical progression :

1. The first term (a).

2. The last term (1).

3. The ratio (r).

4. The number of terms (n).

5. The sum of all the terms (S.).

CASE I.

To find either the First or the Last Term when the other Extreme, the Ratio and the Number of Terms are given.

Ex. 1. The first term of a geometrical series is 2, the ratio 3, and the number of terms 8: what is the last term?

SOLUTION. 37=2187

2 =α

4374

Explanation.-The 2d term equals 1st x 3; the 3d term equals the 1st×3×3, or 1st × 32; the 4th term equals the 1st×3×3×3, or 1st × 33; and so on. X Hence, the 8th term equals 1st × 37, or 2 × 37=4374.

2. The last term of a geometrical series is 4374, the ratio 3, and the number of terms 8: what is the first term?

[blocks in formation]

Explanation. The first term is a number which if multiplied by 37 will produce the 8th term ; hence, if we divide the 8th term by 37 it will pro4374÷37, or 43742187 = 2, the 1st term.

From the foregoing we derive the following

RULES.

1. If the given extreme is the first term, multiply it by the ratio raised to a power one less than is indicated by the number of

terms.

2. If the given extreme is the last term, divide it by the ratio raised to a power one less than is indicated by the number of

terms.

From the foregoing rules we have the following formulas:

[ocr errors][merged small]

3. The first term of a geometrical series is 5, the ratio 2, and the number of terms 6: what is the last term? Ans. 160. 4. The last term of a series is 2187, the ratio 3, and the number of terms 5: what is the 1st term?

5. The last term of a series is 384, the number and the ratio 2: what is the 1st term?

Ans. 27.

of terms 7,

Ans. 6.

6. What is the 6th term of the series 2187, 729, 243, etc.

Ans. 9.

7*. What is the compound amount of $50 for 5 yr., at 6% ? Ans. $66.91+.

8. If a farmer plant a quart of corn, and it produce 9 bu., how much corn will he have at the expiration of 4 yr. if he plant the entire harvest each succeeding spring?

CASE II.

Ans. 214990848 bu.

To find the Sum of a Geometrical Series when the Extremes and the Ratio are given.

Ex. 1. The first term of a series is 6, the last term 162, and the ratio 3: what is the sum of the series?

Ans. 240.

[blocks in formation]

NOTE.-The foregoing statement is derived as follows:

[blocks in formation]

From the above we derive the rule for finding the sum of a geometrical series, as follows:

RULE.

Multiply the last term by the ratio, and from this take the first term, and divide the remainder by the ratio, less one.

The following is the formula: S.

rl-a

r- 1

2. A clerk whose first year's salary is $75 has his salary doubled every year: how much does he earn in 5 years?

Ans. $2325.

3. A merchant engaging in business makes $500 the first year, and continues in business 6 yr., making each year 3 times as much as the preceding year: how much does he make? Ans. $182000.

4. What is the compound amount of $50 for 8 yr. at 6% ?* Ans. $79.6924.

CASE III.

INFINITE SERIES.

In a decreasing infinite series the last term becomes so

small that it may be considered zero; and the formula

α

rx l-a

- 1

r

[merged small][merged small][merged small][merged small][ocr errors][merged small]

hence

r

1

[blocks in formation]

we have the following

RULE.

The sum of an infinite series equals the 1st term, divided by 1

minus the rate.

1. What is the sum of the series 1, 1, §, 27, etc.?

[blocks in formation]

Ans. 33.
Ans. 2.

2. What is the sum of the series 3,,, etc.? 3. What is the sum of the series 2, 1, §, etc.? 4. A dog and a rabbit are 100 yd. apart; if the rabbit run 60 yd. while the dog runs 100, how far will the dog run to catch the rabbit? Ans. 250 yd.

CHAPTER XIV.

ANNUITIES.

314. An Annuity is a sum of money payable at certain periods, usually annually.

315. A Certain Annuity is one which is payable for a certain or definite time.

316. A Perpetual Annuity, or Perpetuity, continues for

ever.

317. A Contingent Annuity is one payable for an uncertain period, usually for a lifetime. It is dependent on some contingent event, such as a death.

318. An Immediate Annuity begins at once; a Deferred Annuity begins at some future time.

319. The Final Value of an annuity is the sum of all the payments, with interest on each from the time it becomes due. 320. The Present Value of an annuity is such a sum of money as will, if put on interest for the given time and at the given rate per cent., amount to the final value; or, in other words, it is the present worth of the final value.

SECTION I.

ANNUITIES AT SIMPLE INTEREST.

All annuities at simple interest may be solved by applying the rules and principles pertaining to Arithmetical Progression and Interest.

CASE I.

To find the Final Value of an Annuity at Simple Interest. Ex. 1. What is the final value of an annuity of $500 for Ans. $6350.

10 yr., at 6% ? SOLUTION.

$500 ×.06 × 9

=

[ocr errors]

Explanation. The 10th payment = $500;

$270 the 9th, $500, with interest for 1 yr., at 6%, or 500 $530; the 8th, $500, with interest for 2 yr., at $770 6%, or $560; and so on to the first, which is ($770+$500) 1o=$6350 $500, with interest for 9 yr., at 6%, or $770; the whole forming an arithmetical series of 10 terms, with $500 for the first term, and $30, the interest for 1 yr., at 6%, for the common difference. By the rule for the sum of an arithmetical series, we have the the first term ($500)+the last term ($770), multiplied by 5, half the number of terms, equal to $6350, the final value of the annuity. The final value may also be found by the rule for deferred payments.

sum =

9 × 10×12

4

x500 $1350

10×500 5000

[ocr errors]

$6350

Explanation. The number of interest-bearing payments or terms is 9; hence the total interest is $1350; and 10 payments of $500 each are $5000; both payment and interest, therefore, equal $1350+$5000, or $6350.

2. What is the final value of an annuity of $400 for 20 yr., at 5% ? Ans. $11800.

3. If a man buy a farm on which he is to pay $300 annually for 15 yr., what is due at the expiration of the time if he fail to make the annual payments, interest at 6% ?

CASE II.

Ans. $6390.

To find the Present Value of an Annuity at Simple Interest. NOTE. Since the present worth of an annuity is the present worth of the final value, it is only necessary to find the final value, as in the preceding case, and then find the present worth of this amount.

1. What is the present worth of an annuity of $1200 for 15 yr., at 6%?

[blocks in formation]

$25560÷1.90=13452.63 = present worth of $25560 due 15 yr. hence,

at 6%.

« ΠροηγούμενηΣυνέχεια »