CASE VII.

862. Given, the sum of the terms and any two of these three, the first term, the last term, or the rate, to find the remaining one.

863. From the formula for the sum we readily obtain the three following formulas, which the pupils will derive and state in the form of rules:

r=

S-a (r−1)S+a; a=tr—(r−1)S.

s-l

=

EXAMPLES FOR PRACTICE.

1. Given, the first term 12, the last term 26244, and sum of the series 39360, to find the rate.

2. Given, the first term, the last term

Ans. 3.

256 6561,

and sum

Ans..

3. Given, the rate 5, the first term 8, and the sum of the

series 156248, to find the last term.

Ans. 125000.

4. The rate is, the first term 200, and the sum of the series 266333; what is the last term?

25

Ans. 512

5. The last term is 196608, the rate 8, and the sum of the series 224694; what is the first term?

Ans. 6.

6. The last term is 13611, the rate, and the sum of the series 368515; ; what is the first term?

Ans. 1024.

864. Since there are five quantities in Geometrical Progression, any three of which being given, the other two may be found, there are twenty distinct cases.

865. The rules for the eight simple cases are expressed

NOTE. For the formulas by which every one of the twenty possible cases may be solved, see Elementary Algebra.

INFINITE SERIES.

866. An Infinite Series is a series in which the number of terms is infinite.

867. In a descending series of an infinite number of terms, the last term becomes so small that it is considered

Rule. To find the sum of an infinite series, divide the first term by unity diminished by the rate.

1. What is the sum of the infinite series 2+3+3+27, etc.?

SOLUTION.-In this series, the first term is

OPERATION.

2, and the rate, and the last term may be regarded as zero, hence the series equals 2 Sum divided by 1-3, or 2÷3, which equals 3.

EXAMPLES FOR PRACTICE.

Find the sum of the following infinite series :

2. Of 1, 1, 1, 1, etc.

3. Of 1, 8, 32, 128, etc.

4. Of 1, 1, 1, 27, etc.

5. Of 3, 1, 2, etc.

45

45

6. Of .45.4545 etc.=1+1000, etc.

7. Of .216 and .4158.

8. Of .35135 and .9285714.

9. 1-1+1+1, etc.

10. If a body should move a mile the 1st second, 4 of a mile the 2d second, and so on until it stops, how far would it move? Ans. 1 mile.

11. A ball dropped from the ceiling of a room 12 ft. high, bounds back 6 ft., then falling, bounds back 3 ft., and so on; how far will it move before coming to rest? Ans. 36 ft.

12. A fox and hound, 16 rods apart, run so that when the hound has run the 16 rods, the fox has run 4 rods, and when the hound has run these 4 rods, the fox has run 1 rod, etc.; how far will the hound run to catch the fox?

Ans. 21 rods.

SECTION XII.

HIGHER PERCENTAGE.

COMPOUND INTEREST.

868. Compound Interest is interest on both principal and interest, when the interest is not paid when due.

Compound interest assumes that if the borrower does not pay the interest when due, it is proper that he should pay interest for it until paid. Some regard it as just, but it has not the sanction of law.

869. Compound Interest, like Simple Interest, may be treated under four cases.

CASE I.

870. Given, the principal, the rate, and the time, to find the compound interest or amount.

1. What is the compound interest of $500 for 3 years, at 5%?

SOLUTION.-Multiplying by the rate per cent., we find the interest for 1 year to be $25; adding this to the principal, we find the amount to be $525, which is the principal for the second year; multiplying the new principal by the rate, we find the interest for the second year to be $26.25, and adding this to the 2d principal, we find the amount for the 2d year to be $551.25; and so proceeding, we find the amount for 3 years to be $578.81, from which we subtract the first principal, and the remainder, $78.81, is the compound interest. Hence the following

Rule.-I. Find the amount of the principal for the first period of the time for which interest is reckoned, and make this the principal for the second period.

II. Find the amount of this principal for the next period; and thus continue till the end of the given time.

III. Subtract the given principal from the last amount, and the result will be the compound interest.

NOTES.-1. When the interest is due semi-annually or quarterly, we find the interest for such time and proceed as above directed.

2. When the time is for years, months, and days, find the amount for the years, then compute the interest on this for the months and days, and add to the last amount before subtracting.

2. What is the compound interest of $650 for 5 years 3 months? Ans. $232.89. 3. What is the compound amount of $5340 for 4 yr. 3 mo. 8 da. at 7% ? Ans. $7133.03. 4. What is the compound interest of $5000 at 10% for 2 years, payable quarterly? 5. What is the amount of $8350 for 5 yr. 7 mo. 24 da. at 8%, payable semi-annually? Ans. $13008.69. 6. Find the compound interest of $1800, invested at 7% for 3 years, and then at 8% for 2 years. Ans. $772.

Ans. $1092.01.

871. The calculation of compound interest is facilitated by the use of a table, for which see Appendix.

Rule. Find from the table the amount for the given number of periods at the given rate, and multiply this amount by the principal. If there is any remaining time, find the amount of this product at the given rate for the time; the result will be the compound amount, from which subtract the given principal for the compound interest.

NOTES.-1. If the time exceeds the limits of the table, calculate the amount for a convenient length of time by the table, take this amount as a principal, and calculate the amount for the remaining time.

2. If partial payments are made on notes bearing compound interest, the amount of the principal must first be found, and the sum of the amounts of the indorsements subtracted from it.

1. What is the compound interest of $7500 for 25 years, at 8%? Ans. $43,863.56. 2. What is the compound interest of $5760 for 15 yr. 4 mo. 24 da., at 10% ? Ans. $19,263.39.

3. What is the amount of $664 for 30 yr. at 6%, payable semi-annually? Ans. $3911.77. 4. What is the compound interest of $100 for 40 years at 8%, interest payable quarterly?" Ans. $2276.98.

5. What is the difference between the simple and compound interest of $400 for 33 yr. 4 mo.? Ans. $1590.96.

6. What sum in 15 yr. 2 mo. 27 da., at 6 per cent. simple interest, will amount to the same as $5000 for the same rate and time at compound interest, payable semi-annually? Ans. $6431.07.

7. A gentleman deposits in a savings bank, at the birth of his son, $1000 to be paid him when he comes of age, interest at 6% compounded semi-annually; what will the deposit amount to at the time it is due ? Ans. $3460.70.

8. Mr. Adams left $20,000 to be equally divided between his son and daughter, directing that the daughter, who was 8 yr. 6 mo. 18 da. old, should receive her share when she was 18 years old, and the son, who was 10 yr. 3 mo. 15 da. old, should receive his when he was 21; what will each receive, if the money is invested in a savings bank at 4 per cent. compounded semi-annually?

9. $700.

Ans. Son, $15282.97; daughter, $14539.55. NEW YORK, MAY 19, 1876. Three months after date, I promise to pay James Wilkins, or order, Seven Hundred Dollars, for value received, with compound interest at 7%.

GEORGE BOOTH.

Indorsements: Dec. 15, 1876, $100; May 19, 1877, $300; Sept. 30, 1877,

$150.

What was due May 19, 1878?

CASE II.

Ans. $213.47.

872. Given, the compound interest or amount, the time, and the rate, to find the principal.

1. What principal, at 6 per cent. compound interest, will yield $1007.26 in 7 years?

OPERATION.

$1007.26.50363 $2000.

SOLUTION.-The compound interest of $1 for 7 years at 6% is $0.50363+, and $1007.26 is the compound interest of as many dollars as $0.50363 is contained times in $1007.26, which is $2000. Rule. Divide the given interest or amount by the interest or amount of $1 for the given rate and time, to find the principal.