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anitri-ET--AL PROGREssion |t.

Rule.—Multiply the sum of the extremes by the number eams, and half the product will be the answer.


1. The urst term of an arithmetical series is 3, the las, term 23, and the number of terms 11 : required the sum of the series. 23-3–26 sum of the extremes. Then 26 x 11-2=1.43 the Answer. 2. How many strokes does the hammer of a clock strike in 12 hours. Ans. 78. 3. A merchant sold 100 yards of cloth, viz. the first ard for 1 ct the second for 2 cts, the third for 3 cts. &c. demand what the cloth came to at that rate 1 Ans. $50. 4. A man bought 19 yards of linen in arithmetical progression, for the first yard he gave Is. and for the last wd. 11, 17s. what did the whole come too - An EIS is. 5. A draper sold 100 yards of broadcloth, at 5 cts, for the first yard, 10cts. for the second, 15 for the third, &c. increasing 5 cents for every yard; what did the whole amount to, and what did it average per yard 7 Ans. Amount $2524, and the average price is so, 52 cts. 5 mills per yard. 6. Suppose 144 oranges were laid 2 yards distant from each other, in a right line, and a basket placed two yards from the first orange, what length of ground will that boy travel over, who gathers them up singly, returning with them cue by one to the basket? Ans. 23 miles, 5 furlongs, 180 yds.


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

Rule.—divide the difference of the extremes by the number as terms less 1, and the quotion will be the common difference,



1. The extremes are 3 and 29, and the number of tenna

14, what is the common difference?
3. } Extremes.

Number of terms less 1=13)25(2 Ans.
2. A man had 9 sons, whose several ages differed alike,

the youngest was three years old, and the oldest 35; what was the common difference of their ages?

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Ans. 4 years.

3. A man is to travel from New-London to a certain place in 9 days, and to go but 3 miles the first day, increasing every day by an equal excess, so that the last day's journey may be 43 miles: Required the daily increase, and the length of the whole journey :

Ans. The daily increase is 5, and the whole journey 207 miles.

4. A debt is to be discharged at 16 different payment, (in arithmetical progression,) the first payment is to be 14t. the last 100l. : What is the common difference, and that sum of the whole debt?

Ans. 5.14s. 8d, common difference, and 912. the whol. debt.


Given the first term, last term, and common difference, to find the number of terms.

Ruur-Divide the difference of the extremes by the common difference, and the quotientincreased by 1 is the number of terms

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1. If the extremes be 3 and 45, and the common differ ence 2; what is the number of terms? Ans. 22.

2. A man going a journey, travelled the first day five miles, the last day 45 miles, and each day increased his journey by 4 miles; how many days did he travel, and how far?

Ans. 11 days, and the whole distance travelled 275 miles

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is when any rank or series of numbers increase by one common multiplier, or decrease by one common divisor as, 1, 2, 4, 8, 16, &c. increase by the multiplier 2; and 27, 9, 3, 1, decrease by the divisor 3.


The first term, the last term (or the extremes) and the ratio given, to find the sum of the series

RULE. Multiply the last term by the ratio, and from the product subtract the first term; then divide the remainder by the ratio, less by 1, and the quotient will be the sum of all the terms. ----------

1. If the series be 2, 6, 18, 54, 162,486, 1458, and the natio 3, what is its sum total! 3× 1-15S.–2 - 3–1 2. The extremes of a geometrical series are 1 and 65536, and the ratio 4 ; what is the sum of the series? Ans. 87:281.

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l'Ivoir LEM1 11,

Given the first term, and the ratio, to find any other term assigned."


When the first term of the series and the ratio are equal."

* As the last term in a long series of numbers is very tedious to be sound by continual multiplications, it will be necessary for the readier noting it out, to have a series of number-inorithmetical proportion, called indices, whose commun diorence is 1.

* When the first term of the series and the ratio are equal, the indices must begin with the unit, and in this case, the product of any two terms is equal to that term, signified by the sum o inutees:

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