Divide the sum of the series by the difference between the sum and the last term; the quotient will give the ratio, and the remainder, after the operation, the first term.*

EXAMPLE.

If the number of terms be 4, the last term 54, and the sum of the series 80; required the first term and the ratio?

From the sum=30

Take the last term=54

Divide by the difference=26)80(3 the ratio.

78

The first term= 2

The following Table exhibits a summary view of the doctrine of Geometrical Progression.

4095

By the second rule, Prob. XIII. it is found that if the sum of the series be di vided by the ratio, the quotient is the difference between the sum of the series and the greatest term, and the remainder is the first term. Therefore, if the sum of the series be divided by the difference between the sum and the last term, the quotient must be the ratio, and the remainder the least term. In the example there taken, the greatest term divided by the ratio, or 1023 with a remain4 der 3 the first term. But as 1023=4095-3072 or the difference between the 4095 sum of the series and the last term, therefore 4095-3072 leaves a remainder 3, the first term. The same must hold true in every series, where the ratio is greater than the least term.

=4, the ratio, and

EXAMPLES IN GEOMETRICAL PROGRESSION,

To be solved by those formulæ in the Table, of which the rules are not given in the Problems.

1. The least term is 2, the greatest term 4374, and the sura of the series 6560; required the number of terms.

This is solved by the 2nd rule of Case 4 in the table.

Ans. 8.

2. If the ratio be 4, the last term 3072, and the sum of the series 4095; what is the first term and the number of terms?

Ans. The first term 3, the number of terms 5,

Wrought by Case 9.

3. Given the number of terms 4, the last term 54, and the sum of the series 80, to find the first term and ratio.

Ans. The first term 2, the ratio 3.

Wrought by Case 9 in the table.

INTEREST.

INTEREST is a premium allowed by the Borrower to the Leader, for the use of his money. It is estimated at a certain number of pounds, dollars, &c. for each hundred pounds, dollars, &c. for a year; and in the same proportion, for a less or greater sum, or for a longer or shorter time. Hence, interest is said to be so much per cent, or per centum, per annum.

The Principal is the sum lent.

*

The Rate is the premium per cent. agreed on."
The Amount is the sum of the principal and interest.
Interest is of two kinds, Simple and Compound.

SIMPLE INTEREST.

Simple Interest is that which is allowed on the Principal only. Note. By this Rule, Commission, Brokerage, Insurance, purchasing Stocks, or any thing else, rated at so much per cent. are calculated.

GENERAL RULE.

1. Multiply the Principal by the Rate, and divide by 100 (or cut off the two right hand figures in the Pounds) and the quotient, or left hand figures will be the answer in Pounds, &c. the right hand figures being reduced and cut off as at first. If the principal be dollars, the right hand figures will be cents.f

2. For more years than one, multiply the Interest of one year by the number of years.

3. For any number of months take the aliquot parts of a year; and for days, the aliquot parts of 30.

1. What is the interest of £573 13s. 91d. at 6 per cent. per annum? Ans. £34 8s. 5d.

* Lawful or legal interest is that which is permitted by the laws of the State. It is different under different governments. In England the Rate is five per cent. In the New England States, it is six, and, in the State of New York it is sever per cent. The courts of the United States allow interest according to the practice of the State where the suit was commenced. The rules of the Courts in the States of Massachusetts, Connecticut, and New York, for computing legal interest, will be given immediately before Discount.

+ This rule is a contraction of a process in the Rule of Three. Thus in the

£ Rate. Principal.

1st Example. As, 100 :: 6 :: £573 13s. 94d.: the interest for a year. And 6X£573 13s. 94d. £34 8s. 5d. the interest. The remainder of the rule is

obvious.

100

The reason of the rule in the Note above is, that as 9 is of 12 months, it the rate be 9 per cent, multiply the principal by of the months; if the rate be 8 cent. by of the number of months; and if 6, by half the months,

&c.

per

£573 13 9

6

£34.42 2 9
20

8.42

12

5.13

4

⚫52

2. What is the interest of £329 176. 61d. for 3 years, 7 months, and 12 days, at 5 per cent. per annum. Ans. £59 13s.

£329 17 61d.

Then,

3. What is the interest of 347 dollars 50 cents, at 6 per cent.

per annum for a year.

$347-50 Or $100: 6: 347.50c.: $20 85c.

6

20-8500 Ans. $20 35c.

4. What is the interest of $797 13c. at 6 per cent. per annum, for 8 months?