1

Ex. 2. 2,

Common Ratio.

Here Common Ratio:

&c. are in Geometrical Progression, find the 10'

=

288. If any terms be taken at equal intervals in a geometrical progression, they will be in geometrical progression.

2n

3n

Let a, ar...ar”.............. ɑ p21.... a p31....&c. be the progression, then a, ar", ar2n, ar", &c. are at the interval of n terms, and form a geometrical progression, whose common ratio is ï”.

289. If the two extremes, and the number of terms, in a geometrical progression be given, the means, that is, the intervening terms, may be found.

Let a and 7 be the extremes, n the number of terms, and r the common ratio; then the progression is a, ar, ar2, ar3.....ar2-1; and since is the last term,

and

being thus known, all the means, or intervening terms, ar, ar2, ar3, &c. are known.

COR. A single Geometrical mean between a and b, called the

Geometrical mean, will be a (2) or √ab; but this may be shewn more

simply thus,

Let a, a, b be in Geometrical Progression, theh by Def.

.. x2=ab, and x = Jab.

Ex. There are three means, or intervening terms, in a Geometrical Progression between 2 and 32; find them.

290. To find the sum of a series of quantities in Geometrical Progression.

Let a be the first term, r the common ratio, n the number of terms, and s the sum of the series:

COR. 1. If be the last term, l = ar"-1,

.. 8 =

rl - a*

r- 1

from which equation, any three of the quantities s, r, l, a, being given, the fourth may be found.

COR. 2. When is a proper fraction, as n increases, the value of TM, or of ar”, decreases, and when n is increased without limit, ar" becomes less, with respect to a, than any magnitude that can be assigned t. Hence the sum of the series, which in general is equal

a

This quantity

1

which we call the sum of the series, is the

limit to which the sum of the terms approaches, but never actually attains; it is however the true representative of the series con

* The following is another method of arriving at the same result equally simple :— Let a, b, c, d, &c., h, k, l, be the series, s the sum, and the common ratio; then, by definition,

b=ar, c=br, d= cr, k = hr, l = kr,

...

3 10

†Thus, if r =· or 0.3, r2 = 0·09, 73 = 0·0027; and so on, shewing that as n increases

y decreases, and that such a power of r may be taken as to produce a quantity less than any number which shall be named, however small.

That is, although no definite number of terms will amount to

a

yet, by taking a

1

sufficient number, the sum will reach as near as we please to it; and, whatever number be taken, their sum will not exceed it.

tinued sine fine, for this series arises from the division of a by 1-r;

and therefore

a

1

r

may without error be substituted for it.

Ex. 1. To find the sum of 20 terms of the series, 1, 2, 4, 8, &c.

Ex. 2. Required the sum of 12 terms of the series, 64, 16, 4, &c.

4

-

Required the sum of 12 terms of the series 1, − 3,

291. Recurring decimals are made up of quantities in geo

according as one, two, three, &c. figures recur; and the vulgar fraction, corresponding to such a decimal, is found by summing the series.

Ex. 1. Required the vulgar fraction which is equivalent to the decimal 123123123 &c.

Let 123123123 &c. = s; then, as in Art. 290, multiply both sides by 1000; and 123-123123123 &c. = 1000s, and by subtracting the former equation from the latter, 123 = 9998;

Ex. 2. Required the vulgar fraction which is equivalent to PPP &c. where P contains p digits recurring in inf.

Ex. 3. Required the vulgar fraction equivalent to PQQQ &c., where P contains p digits, and Q contains q digits recurring in inf.

Both these results may be easily verified by expressing the proposed quantities in geometric progressions-the former being

which may be summed by the rule in Art. 290, Cor. 2.

or <,

1

292. In a Geometrical series continued in inf. any term the sum of all that follow, according as the common ratio <,=, or >

Let a + ar+ar2 + ... ar2-1 + ar2 + the series after n terms is the sum of

......

2

be the series. Then the sum of

HARMONICAL PROGRESSION.

DEF. Any magnitudes A, B, C, D, E, &c. are said to be in Harmonical Progression, if A: C: A - B : B-C; B: D: B-C: CD; C: E:: C-D: D-E; &c.

293. The reciprocals of quantities in Harmonical Progression are in Arithmetical Progression.

Let A, B, C, &c. be in Harmonical Progression ;

then by Def. A CA-B: B-C;

.. AB AC AC - BC, (Art. 237),

=