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RULE. Multiply the sun of the extremes by the num) terms, and half the product will be the answer.

EXAMPLES. 1. The first term of an arithmetical series is 3, th term 23, and the number of terms 11; required th of the series. :23+3=26 sum of the extremes.

Then 26x11-2–143 the Answer. 2. How many strokes does the hammer of a trike, in twelve hours ?

Ans. -3. A merchant sold 100 yards of cloth, viz. the yard for 1 ct. the second for 2 cts. the third for 3 cts I demand what the cloth came to at that rate ?

Ans. $50 4. A man bought 19 yards of linen in arithmetical gression, for the first yard he gave 18. and for the las il. 178. what did the whole come to P Ans. £18 1

5. A draper sold 100 yards of broadcloth, at 5 cts the first yard, 10 cts. for the second, 15 for the third, increasing 5 cents for every yard ; what did the w amount to, and what did it average per yard ?

Ans. Antount $2521, and the average price is $2,5% 5 mills per yard.

6. Suppose 144 oranges were laid 2 yards distant fi each other, in a right line, and a basket placed two ya from the first orange, what length of ground will that i travel over, who gathers them up singly, returning v them one by one to the basket ?

Ans. 23 miles, 5 furlongs, 180 yds

PROBLEM II. The first term, the last term, and the number of ter given, to find the common difference.

RULE. Divide the difference of the extremes hy the numi of terms less l, and the quotient will be the commond

ference. ::

::.. EXAMPLES 1. The extremes are 3 and 20, and the number of terms 14, what is the common difference?

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29 Extremes. : .-3

Number of terms less 1=13)26(2 Ans. c. 2. A man had 9 sons, whose several ages differed alike,

the youngest was 3 years old, and the oldest 35; what was the common difference of their ages ?

Ains. 4 years 3. A man is to travel from New-London to a certain place in 9 days, and to go but s 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 id

crease, and the length of the whole journey? d. Ans. The daily increase is 5, and the whole journey

207 miles. . 4. A debt is to be discharged at 16 different payments (in arithmetical progression, the first payment is to be 141. the last 100l. : What is the common difference, and the sum of the whole debt?

Ans. 51. 14s. 8d. common difference, and 9121. the whole

debt.

PROBLEM III. Giver the first term, Tast term, and common difference, to

find the number of terms.

RULE. Divide the difference of the extremes by the common difference, and the quotient increased by 1 is the number of terms.

EXAMPLES. 1. If the extremes be 3 and 45, and the common difference 2; what is the nunber 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: ins. 11 days, and the whole distance travelled 275 miles, GEOMETRICAL PROGRESSION, Is when any rank or series of numbers increased by common multiplier, or decreased by one common divi as 1, 2, 4, 8, 16, &c. increase by the multiplier 2; 27, 9, 3, 1, decrease by the divisor 3.

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

RULE. Multiply the last term by the ratio, and from the duct subtract the first term; then divide the remail by the ratio, less by 1, and the quotient will be the : of all the terms.

EXAMPLES. 1. If the series be 2, 6, 18, 54, 162, the ratio 3, what is its sum total ? :::::3x1458-2

-E2186 the Answer.

S- 1 2. The extremes of a geometrical series are 1 a 65536, and the ratio 4; what is the sum of the series :

Ans. 87381. PROBLEM II. Given the first term, and the ratio, to find any other ter

assigned. *

CASE I. When the first term of the series and the ratio are equal

**As the last term in a long series of numbers is very i dicus to be found by continual multiplications, it will necessary for the readier finding it out, to have a seri of numbers in arithmetical proportion, called indice whose common difference is 1.

+1Vhen the first tern of the series and the ratio are equo the indices must begin with the unit, and in this case, ti

1. Write down a few of the leading terms of the siries, and place their indices over them, beginning the indices with an unit or l..."

2. Add together such indices, whose sum shall make up the entire index to the sam required.

3. Multiply the terms of the geometrical series belonging to those indices together, and the product will be the term sought:

EXAMPLES 1. If the first be 2, and the ratio 2; what is the 15th term. 1, 2, 3, 4, 5, indices. Then 5+5+3=13 2, 4, 8, 16, 32, leading terms. 32X32X8=8192 Ans.

2. A draper sold 20 yards of superfine cloth, the first yard for 3d, the second for 9d. the third for 27d. &c. in triple proportion geometrical ; what did the cloth come to at that rate?

The 20th, or last term is 5486784401d. Then 3+3486784401.-3

- =5230176600d. the sum of all

9- 1 the terms (by Prob. I.) equal to £21792402 10s. Ans.

3. A rich miser thought 20 guineas a price too much for 12 fine horses, but agreed to give 4 cents for the first, 16 cents for the second, and 64 cents for the third horse, and so on in quadruple or fourfold proportion to the last: what did they come to at that rate, and how much did they cost per head, one with another?

Ans. The 12 horses came to $223696, 20cts. and the average price was $18641, 35cts. per head.

produet of any two terms is equal to that term, signified by the sum of their indices.

S1 2 3 4 5 &c. Indices or arithmetical series. se { 2 4 8 16 32 &c. geometrical series. , Mam 3+2 = 5 = the index of the fifth term, and Weus 4x6 = 32 = the fifth term. ,

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. . CASE II
When the first term of the series and the ratio are diffe-

rent, that is, when the first term is either greater ur
less than the ratio.*

1. Write down a few of the lealling te;'nds er the series,
and begin the indices with a cypher: Thus, 0, 1, 2, 3, &c.
:.2. Add together the most convenient indices to make
an index less by 1 than the number expressing the place
of the term sought."

3. Multiply the terms of the geometrical series to-
gether belonging to those indices, and make the product
a dividend.

4. Raise the first term to a power wliose index is one
less than the number of the terms multiplied, and make
the resulta divisor.
5. Divide, and the quotient is the term sought.

EXAMPLES.
4. If the first of a geometrical series be 4, and the rati*
S, what is the 7th term
..0, 1, 2, 3, Indices.

4, 12, 36, 108, leading terms.

3+2+1=6, the index of the 7th term..
108 X36X12=46656
.:

2916 the nth term required ::

16
Here the number of terms multiplied are three; there.
fore the first term raised to a power less than three, is the
ed power or square of 4=16 the divisor.

* When the first term of the series and the ratio are dif.
ferent, the indices must begin with a cypher, and the sum
of the indices made choice of uust be one less than the num-
ber of terms given in the question: because 1 in the indices
stands over the second term, and 2 in the indices over the
third term, &'c. and in this case, the product of any tuo
terins, divided by the first, is equal to that term beyond the
first, signified by the sum of their indices.
Thus. So, 1, 2, 3, 4, ģc. Indices.

nous, 21, 3, 9, 27, 81, c. Geometrical series.

Here 4+3=7 the index of the 8th term. 1. 81X27=2187 the 8th term, or the 7th beyond the 1st

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