Hence we derive the following general rule, the correctness of which is evident, whatever be the first term, or ratio.

RULE.

Raise the ratio to the power whose index is one less than the number of terms given, which multiply by the first term; that product is the last term, or greater extreme.

2. Multiply the last term by the ratio, and from the product subtract the first term, and divide the remainder by the ratio less one; the quotient will be the sum of the series.

Or, raise the ratio to a power equal to the number of terms; subtract one from that power; multiply the remainder by the first term; divide this product by the ratio, less one; the quotient will be the sum of a geometrical series.

1. The first term is 3, and the ratio 2; what is the 6th term? 2×2×2×2×2=25=32; 32 × 3, 1st term, =96. Ans.

Having given the ratio and the two extremes, to find the sum of the series.

RULE.

Subtract the less extreme from the greater; divide the remainder by one less than the ratio, and to the quotient add the greater extreme; the sum will be the sum of the series.

2. The first term is 3, and the ratio 2, and the last term 192; what is the sum of the series? 192-3-189, diff. of extremes, 2-1=1)189(189; then 189+192-381. Ans.

3. If the first term be 2, and the ratio 2, what is the 13th term? 1, 2, 3, 4, 5× 5×3=13, or, 2×212 (less, 1)=8192. 2, 4, 8, 16, 32 x 32 x 8=8192. Ans.

4. If the first term be 5, and the ratio 3, what is the 7th term?

0, 1, 2, 3,+2 +1 =6=indices to 6th t'm beyond 1st or 7th. 5, 15, 45, 135,X45X15=91125 dividend.

The number of terms multiplied is three, namely: 135 × 45 ×15, and 3-1-2, is the power to which the term 5 is to be raised; but the second power of 5 is 5 x5=25, therefore, 91125-25=3645=7, term required.

NOTE. The following examples will embrace the general operation of the several rules of geometrical progression, sufficient for common use.

5. A. purchased 24 yards of sheeting, for which he paid 2c. for the first yard, 4c. for the second, 8c. for the third, &c., increasing in a duplicate proportion; required the amount.

2-1=1)335544.30 Ans. in dollars and cents.

6. What debt can be discharged in a year, by paying 1 cen the first month, 10 cents the second, and so on, each morth in a tenfold proportion?

1

2

3

4

5

10 x 100 x 1000 × 10000 × 100000 × 1000000

6

1000000

1000000000000

1

999999999999

1 x

10-19)999999999999

D1111111111.11 Ans.

7. What will one cent amount to, if you double it every year

Ans. D20971.51.

for 21 years? 8. The first term of a series having 10 terms, is 4, and the ratio 3, what is the last term?

Ans. 78732 9. What is the last term of a series having 18 terms, the first of which is 3, and the ratio 3 ? Ans. 387420489.

10. A man was to travel to a certain place in 4 days, and to travel at whatever rate he pleased; the first day he went 2 miles, the second 6 miles, and so on to the last, in a threefold ratio; how far did he travel the last day, and how far in all?

Ans. last day, 54 miles; in all, 80 miles

11. Sold 14 pairs of stockings, the first at 4 cents, the second at 12 cents, and so on in a geometrical progression; what did the last pair bring him, and what did the whole bring him?

Ans. last, D63772.92; whole, D95659.36.

12. If our ancestors who landed at Plymouth, A. D. 1620, being 101 in number, had increased so as to double their number in every 20 years, how great would have been their population at the end of 1840 ? Ans. 206747. 13. A sum of money is to be divided among 10 persons; the first to have D10, the second, D30, and so on, in a threefold proportion; what will the last have? Ans D196830. 14. A man bought a horse, and by agreement was to give lc. for the first nail, 2c. for the second, 4c. for the third, &c. ; there were 4 shoes, and 8 nails in each shoe; what was the cost of the horse? Ans. D42949672.95.

REVIEW.

What is Geometrical Progression? How do you form a geometrical progression ? What is the common ratio? What is the difference between arithmetical and geometrical progression? What is an ascending series? What is a descending series? What are the several numbers called? In every geometrical progression how many things are to be considered? What are they? What are the first and last terms called?

What are the intermediate terms called?

15. If the ratio be 4, the number of terms 6, and the greatest term 3072; what is the sum of the series?

Ans. 4095.

Divide the last term by the 5th power of the ratio, &c.

PERMUTATION.

PERMUTATION is the method of finding how many different ways any number of things may be changed. Thus take the first three letters of the alphabet, a b c; they will admit of six changes, a b c, a c b, ba c, b c a, c ba, cab, and so on, according to the given number of terms.

RULE.

Multiply all the terms of the natural series constantly from 1, or unity, to the given number, inclusive; the last product will be the number of changes required.

1. In how many different positions can five persons be placed at a table? Thus 1×2×3×4×5=120. Ans.

:

2. What time will it require for 8 persons to seat themselves differently every day at dinner? Ans. 110 years, 1421 days. 3. How many variations can be made of the English alphabet, it consisting of 26 letters?

Ans. 403291461126605635584000000. 4. How many changes may be rung on 15 bells; and in what time may they be rung, allowing three seconds to every round? Ans. 1307674368000 changes; 3923023104000 seconds. 5. How many variations may there be in the position of the nine digits? Ans. 362880. 6. A man bought 25 cows, agreeing to pay for them 1 cent for every different order in which they could all be placed; how much did the cows cost him?

Ans. D155112100433309859840000. 7. Christ's church, in Boston, has 8 bells; how many changes may be rung upon them. Ans. 40320.

COMBINATION.

COMBINATION teaches how many different ways a less number of things may be combined out of a larger; thus out of the letters a b c d, are six different combinations of two, namely, a b, a c, a d, d c, d b, b c.

Thus: 4×3=12; 1×2=2)12(=6. Ans.

RULE.

Take a series, proceeding from and increasing by a unit, up to the number to be combined; then take a series of as many places, decreasing by unity, from the number out of which the combinations are to be made; multiply the first continually for a divisor, and the other for a dividend; the quotient will be the

nswer.*

1. How many combinations of five letters in ten?

Thus 10×9×8×7×6=30240, dividend; 1×2×3×4x5 =120, divisor.

Ans. 252. 2. How many combinations of ten figures may be made out of twenty? Ans. 184756.

3. How many combinations may be made of seven letters out of twelve.

Ans. 792.

4. How many combinations can be made of six letters out of the 24 letters of the alphabet? Ans. 134596.

REVIEW.

What is Permutation? What is the rule? bination? What is the rule?

What is com

5. How many changes may be rung with 4 bells out of 8? Ans. 1680.*

6. How many variations may be made of the letters in the word Zaphnathpaaneah (15)? 2×6×1×2×120=2880 diviAns. 454053600. 7. How many different numbers can be made of the following figures: 1223334444? Ans. 12600.

sor.

COMPOUND INTEREST, BY DECIMALS.

COMPOUND INTEREST is that which arises from interest and principal added together annually, as the interest becomes due, by the continued multiplication of the new principal by the ratic or rate per cent. ; thus, if I owe A. D100, payable on demand, and neglect to pay either interest or principal for several years, he would be justified in adding the interest to the principal annually, and computing the interest on this amount: D100+6

106, first year; then D106×6=6.36 interest second year, which D6.00+6.36D.=12.36 interest or D112.36 amount for the second year; D112.36×6=6.74 interest for the third year, which D6.00 +6.36+6.7419.10 interest for the third year, or D119.10.160 amount at compound interest for three years at 6 per cent. In many cases it is considered illegal to receive compound interest, therefore it is seldom computed; but when a note, bond, or obligation, is given with a credit of several years, on condition that the interest shall be paid annually, if the interest is not paid until the obligation becomes due, it is no more than just and right that compound interest should be paid, as well as principal; for the person having the use of the principal has likewise the use of the interest, which of right belongs to another; therefore interest should be computed accordingly.

The following table is computed by the continual multiplication of 1, or D100 by the ratio as above; which may sometimes be found useful where the higher powers are required, or o test the accuracy of calculations.