is (units) greater or less than the common difference, and you have the sum required.

NOTE. When the first term is 1, and the common difference 1, the sum of course will be half of that found by the first part of the rule; and when the first term is 1, and the common difference is 2, the sum will be the square of the number of terms.

1. Bought 15 yards of linen, at the rate of 2 cents for the first yard, 4 for the second, 6 for the third, &c.; what was the cost of the whole? Ans. $2.40.

Here we have only to recollect that the square of 15 is 225, and that if 15 be added to the square the sum will be 240, and we obtain the answer by a very simple mental operation.

NOTE. We will again suggest to the learner never to use figures to solve a question, if he can possibly effect the solution by a purely mental operation, as by pursuing this course the reasoning and retentive powers of the mind will be constantly expanding and improving, whilst at the same time his progress in the acquisition of knowledge will necessarily be much more rapid than that of one who occupies so large a portion of his time, as is generally taken up in the mere mechanical process of making figures.

2. 25 persons agree to contribute to a charitable institution, in the following proportions, viz., the first is to give $2, the second $4, the third $6, and so on; what will be the whole sum contributed? Ans. $650. 3. The clocks of Venice are said to go to 24 o'clock; how many strokes must such a clock strike in 24 hours? Ans. 300 strokes.

4. How many strokes does a common clock strike in 12 hours? Ans. 78 strokes. 5. A farmer offers to sell 85 sheep, at the rate of 5 cents for the first sheep, 10 cents for the second, 15 cents for the third, and so on, increasing 5 cents on the price of each; what will he get for the whole at this rate? Ans. $182.75.

6. A butcher bought 48 oxen, and agreed to pay $1 for the first, $3 for the second, $5 for the third, and so on; what did he pay at this rate for the whole ? Ans. $2304.

7. A man bought 50 yards of cloth, for which he was to pay 6 cents for the first yard, 9 cents for the second, 12 cents

for the third, and so on, increasing by the common difference 3; how much did he pay for the whole? Ans. $39.75.

8. 16 persons gave charity to a poor man, the first gave him 5 cents, the second 9, and so on, increasing by the common difference 4; how much did he received in all ?

Ans. $5.60.

9. A merchant bought 36 yards of cloth, and gave for the first yard 8 cents, for the second 12 cents, and so on, increasing by the common difference 4; how much did he pay for the whole? Ans. $28.08.

10. Suppose 100 stones are placed in a right line 3 yards distant from each other, and the first 3 yards from a basket; what distance must a person travel to gather them singly into the basket? Ans. 17 miles, 380 yds.

11. A new road is to be marked out by 1000 stakes, to be driven into the ground at the distance of 22 feet from each other; the stakes are placed in a pile 22 feet from the point where the first one is to stand. Now suppose a man to start from this point and to take from the pile one stake at a time, and drive it into the ground, and continue the operation until he has driven the whole number in a direct line at the proper distances; what length of time would he require to complete the job, supposing him to walk 39 miles a day upon an average? Ans. 10617 days.

12. A merchant sold 1000 yards of linen, at 2 pins for the first yard, 4 for the second, and 6 for the third, &c., increasing two pins every yard; how much did the linen produce when the pins were afterwards sold at 12 for a farthing? Ans. £86 17s. 101d.

13. If a number of dollars were laid in a straight line for the space of a mile, a yard distant from each other, and the first a yard from a chest, what distance would a man travel who, starting from the chest, should pick them up singly, returning with them one by one to the chest?

Ans. 1761 miles. (0.)

14. Suppose a number of stones were laid a rod distant from each other, for a distance of thirty miles, and the first stone a rod from a basket, what distance will a man travel who gathers them up singly, returning with them one by one to the basket? Ans. 288,090 miles 2 rods.

NOTE. The entire distance from the basket to the furthest stone, will be one rod more than thirty miles. The solution requires 74 figures by the common method; our solution requires but 15.

15. A debt is to be paid at 16 different payments in arithmetical progression; the first payment to be $14, and the last $100; what is the common difference of the payments? Ans. $5.733.

NOTE,—The common difference is equal to the quotient of the difference of the extremes divided by the number of terms less one.

16. A man agrees to travel from Philadelphia to a certain place in 16 days, and to go but 4 miles the first day, increasing every day by an equal excess, so that the last day's journey may be 79 miles, required the daily increase, and what is the whole distance? S Daily increase 5 miles, and whole distance 664.

Ans.

17. A debt is to be paid at 8 different payments in arithmetical progression, the first payment to be $21, and the last $175, what was the common difference, and what is the whole sum? Common difference, $22,

Ans. {and whole sum,

$784.

18. A man going a journey travelled 8 miles the first day, and increased his journey 4 miles every day, until the last day's journey was 56 miles; how many days did he travel? Ans. 13 days.

NOTE. Since the difference of the extremes divided by the number of terms less one gives the common difference, it follows that the difference of the extremes divided by the common difference will give the number of terms less one, to which if 1 one be added we have the number of terms.

19. A farmer sold a quantity of oats at the rate of 4 cents for the first bushel, and the price of every bushel after the first was to increase by the common difference of 3 cents, until the price of the last bushel should be 76 cents; how many bushels did he sell, and what was the average price per bushel? S Sold 25 bushels. Average price 40 cents a bushel.

Ans.

GEOMETRICAL PROGRESSION.

Any series of numbers, the terms of which increase by a common multiplier, or decrease by a common divisor, are said to be in geometrical progression.

As 2, 4, 8, 16, 32, 64, 128, &c.

Or 128, 64, 32, 16, 8, 4, 2.

The former is called an ascending series, and the latter a descending series.

The multiplier or divisor by which the series is increased or decreased is called the ratio.

The first and last terms of the series are called the extremes, and the other term the means.

In any geometrical series, the second term is found by multiplying the first term by the ratio, the third term by multiplying the second term by the ratio; the fourth term by multiplying the third term by the ratio, &c. Hence any term in an increasing series, may be found by multiplying the first term by that power of the ratio which is denoted by the number of terms less one, for instance the fifth term in the series, 2, 6, 18, &c., will be found by multiplying the fourth power of 3 (the ratio) by the first term, (2) thus, 34x2=162.

2, 6, 18, 54,

Again, take the series,
And multiply each term by the ratio, 6, 18, 54, 162.

Here the last term in the second series, that is, (162) is the result of multiplying the last term in the first series by the ratio; now if we subtract the first series from the second, the remainder will be twice the first, because the second is three times the first; if we divide by 2, therefore, we get the sum of the given series-and we see that to subtract the first from the second, we have only to diminish the last term in the second by (2) (the first term of the first.) Hence the

RULE.

Raise the ratio to the power whose index is one less than the number of terms which multiply by the first term, the

product will be the last or greater extreme 2 multiply the fast term by the ratio, from the product subtract the first term, and divide the remainder by the ratio less one for 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, and divide the product by the ratio less one for the sum of the series.

1. Sold 14 yards of broad cloth, at 2 cents for the first yard, 6 cents for the second, 18 cents for the third, and so on in geometrical progression; what did the whole amount to? Here the ratio is 3.

The square or second power of 81 is 6561-8th power, which multiply by 729 (6th power,) 6561×927 will give the 14th power (47829.69) from which if we subtract 1, and multiply the remainder by the first term, and divide the product by the ratio less one, the result will be the sum of the series or answer to the question, but as the first term is 2 and the ratio less one (3-1) is also (2,) we have the answer, $47,829.68, simply by subtracting 1 from the 14th power of the ratio.

2. A thresher wrought 20 days, and was to receive 4 grains of wheat for the first day's labour, 12 for the second, 36 for the third, &c. ; how much did his wages amount to, allowing 7680 grains to make a pint, and the whole disposed of at $1 a bushel?

Here 3 is the ratio, the 4th power of which is 81, and the second power of 81 gives the 8th power of 3 (the ratio)=6561; and the second power of (6561) gives the 16th power of the ratio, which being multiplied by 81 (the 4th power) gives the (20th.)

43046721×81=16th power. 3840)348678440|0=20th power less 1. 90S016+pints=14187 bushels, which

at one dollar per bushel amounts to $14187.75.