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duct of any two terms, equally diftant from

the extremes.

Let a be the firft term, y the laft, r the common ratio; then the feries is,

a, ar, ar2, ar3, art, &c.

1, 2, 2, 2, 2, &c.

It is obvious that any term in the upper rank is equally diftant from the beginning, as that below it from the end; and the product of any two fuch is equal to ar, the product of the first and last.

Prop. II. The fum of a geometrical series wanting the firft term, is equal to the fum of all but the laft term, multiplied by the

common ratio.

For, affuming the preceding notation of a feries, it is plain that

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=rxa+ar+ar2,&c..+1⁄2 + 1⁄2 + 1⁄2 + 1⁄2



Cor. 1. Therefore s being the fum of the

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Hences can be found from a, y, and r; and any three of the four being given, the fourth may be found.

Cor. 2. Since the exponent of r in any term is equal to the number of terms preceding it; hence, in the last term, its exponent will be n-1; the last term, therefore

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of these four, s, a, r, n, any three being given, the fourth may be found by the folution of equations. If n is not a small number, the cafes of this problem will be moft conveniently folved by logarithms; and of fuch folutions there are examples in the Appendix to this part.

Cor. 3. If the feries decreafes, and the number of terms is infinite, then according to this notation, a, the least term, will be o, yr

and s

a finite fum.


Ex. Required the fum of the feries 1,

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What are called in arithmetic, repeating and circulating decimals, are truly geometrical decreasing feriefes, and therefore may be fummed by this rule.

Thus.333,&c. = 3

10 100

+ &c. is a ge

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It was observed, (Chap. I. and IV.) that in many cases, if the divifion and evolution of compound quantities be actually per


formed, the quotients and roots can only be expreffed by the series of terms, which may be continued ad infinitum. By comparing a few of the first terms, the law of the progreffion of fuch a feries will frequently be discovered, by which, it may be continued without any further operation.

When this cannot be done, the work is much facilitated by feveral methods; the chief of which is that by the binomial theo



Any binomial (as a+b) may be raised to any power (m) by the following rules.

1. From infpecting a table of the powers of a binomial obtained by multiplication, it appears that the terms, without their coeffi

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2. The coefficients of these terms will be

found by the following rule.

Divide the exponent of a in any term by the exponent of b increafed by 1, and the 'quotient multiplied by the coefficient of that term will give the coefficient of the next following term.

This rule is found, upon trial in the table of powers, to hold univerfally. The coefficient of the first term is always I; and by applying the general rule now propofed, the coefficients of the terms in order will be MI m-2


as follows, 1, m, m×”—, m mX 2


&c. They may be more conveniently ex


preffed thus, I, Am, B× CX

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m3 &c. the capitals denoting the



preceding coefficient. Hence a+bm =a+

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b3, &c. This is the celebrated binomi

al theorem. It is deduced here by induction only, but it may be rigidly demonftrated, though upon principles which do not belong to this place.


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