ARITHMETICAL PROGRESSION.

Arithmetical Progression is a series of numbers which increase or decrease by a continual addition or subtraction of the same numbers, as 1, 2, 3, 4, 5, 6; 1, 3, 5, 7, 9, 11; 6, 5, 4, 3, 2, 1; 11, 9, 7, 5, 3, 1. There are five things to be particularly attended to in Arithmetical Progression, the first term, the last term, the number of terms, the common difference, and the sum of all the terms.

Case 1.

The first term, common difference, and number of terms being given to find the last term and sum of all the terms.

1. Multiply the number of terms less 1 by the com. mon difference and to that product add the first terni the sum is the last term,

2. Add the first and last terms together, and multiply the sum by the number of terms, and half the product will be the sum of all the terms.

Case 2.

When the first and last terms (or two extremes,) are given to find the common difference.

Rule.

Divide the difference of the extremes by the number of terms less 1, the quotient will be the common difference.

Questions.

What is Arithmetical Progression?

Name the five things which should be particularly attended to in Arithmetical Progression.

By what rule do you work when the first term, common difference, and number of terms are given to find. the last term, and sum of all the terms?

By what rule do you work when the first and last terms are given to find the common difference?

Examples.

1. What is the last term and the sum of all the terms of an Arithmetical Progression whose first term is 1, the common difference 2, and number of terms 192

19-1-18

1+37-38

Sum of all the terms 361 Ans.

2. A person sold 40 yards of linen at 2 cents for the first yard, 4 cents for the second, increasing 2 cents every yard, what did they amount to? Ans. $16.40.

3. A man on a journey, travels the first day 10 miles, the second 14 miles, increasing 4 miles every day, how many miles did he travel the tenth day, and how many miles did he travel in all? Ans. 56 miles 10th day, travelled in all 280 miles.

4. A butcher bought 75 sheep, and gave 6 cents for the first, 8 for the second, &c. what did he give for the the last, and what did the whole number cosť him?

Ans. For the last $1.54, the whole $120.00.

Case 2.

1. If the ages of 12 person are equally different, the youngest is 18 years, and the eldest 40, what is the common difference of their ages?

40

18

12-1=11)22(2 common difference.

22

2. When a debt is paid at 8 different payments in Arithmetical Progression, the first payment to be 21 dol

lars, the last 175 dollars, what is the common difference, and what each payment, and what was the whole debt? Ans. Common difference $22, 2d payment $42,

3d payment $65, &c. whole sum $780. 3. A man received charity form 10 different persons, the first gave him 4 cents, the last 49 cents, in Arithmetical Progression, what was the common difference, and what did the man receive? Ans. Received $2.65. Common difference 5 cents,

GEOMETRICAL PROGRESSION.

Geometrical Progression is the increase of a series of numbers by a common multiplier or decrease by a common divisor as, 2, 4, 8, 16, 32.-32, 16, 8, 4, 2. The ratio is the number by which the series increases or decreases.

To find the last term and sum of the series.

Rule.

Raise the ratio to the power whose index is one less than the number of terms given.

2. Multiply the product by the first term and that product will be the last term.

3. Multiply the last term by the ratio, from the product subtract the first term and divide the remain. der by the ratio less one, for the sum of the series.

Questions.

What is Geometrical Progression?

What is the ratio?

By what rule do you proceed to find the last term and sum of all the series?

Examples.

1. If I buy 16 córds of wood, and agree to pay 2 cents for the first, 4 cents for the second, 8 for the third, &c. doubling the price to the last, what will it cost me ?

Sum of series $1310.70 Ans.

2. A person dying left 8 children, to whom he bequeathed in the following manner, viz. the youngest child to have 57. the next youngest 157. and so on, every child to exceed the next younger in triple proportion, what will be the share of the eldest? Ans. 169001.

3. A person at the birth of his son deposited in bank 1 cent towards his portion, promising to double it at the return of every birth day until he was 21 years of age, what was his portion? Ans. $20971 51 cts.

COMPOUND INTEREST BY DECIMALS.

The ratio in Compound Interest is the amount of 1 dollar or pound for 1 year, which is found as follows, As 100 1 :: 106: 1.06

Note. The 4th root of the ratio will be the quar. terly amount; the square root the half yearly amount; and the product arising from the half yearly and quar. ter yearly multiplied together the 3 quarter yearly amount, as follows

4/1.36 1.007417 quarterly amount: 1.03-1.014889 half yearly amount.

1.007417 x 1.014889-1.022416 amunt for 3 quarters. Note 2.-The 4th root is found by extracting the square root of the square root.

The ratio involved to the power whose index is the time is the amount of 1 dollar or pound for that time as a square for 2 years a cube for 3 years. &c. 1.06 x 1.06 x 1.06-1 1910160 amount of 1 pound or dollar for 3 years.

When the ratio is to be involved to years and quarters the power for the years must be multiplied by the quarterly amount

1.1910160 x 1.004417 1.29769875 for 31 years.

The power of 1 dollar or pound may also be obtained for months and days nearly, by adding the monthly simple interest of 1 pound or dollar, or proper parts thereof to the amount of the quarter next preceding the given time for what that time exceeds the said quarter as follows Amt. for year 1.029563

Int. of $1, for 1 mo. .005000 & for 5 days

..000833

for 43 years

for 1 month

1.311873

.005000

for 7 mo. 5 days

Rate

her

cent.

3

1.034396 Amt. for 4 10 5 1.324706

TABLE I.

Amount of 11. or dollar for a year

anil for grs, at Compound Interest.

Simple

interest

of 11 for

Ratio For 3 grs. |For 2qrs. For 1 qr.] month.

.002500

.002917

1.03 1.022416 1.014889 1.007417 31.035 1.026173 1.017349 1.008637 4 1.04 1.029852 1.019804 1.009853 .003333 41.045 1.033563|1.022252|1.011065| .003750

5

77 1.07 1.052053 1.034408 |1.017058 | .005833