If the number of terms be odd, and the middle term is one of those which are given, then, from what has been proved, the middle term multiplied by the whole number of terms will be the sum required.

Here the 3d and 4th terms are equidistant from the beginning and end;

GEOMETRICAL PROGRESSION.

273. Quantities are said to be in Geometrical Progression, or continual proportion, when the first is to the second as the second to the third, and as the third to the fourth, &c. that is, when every succeeding term is a certain multiple, or part, of the preceding term.

If a be the first term, and ar the second, the series will be a, ar, ar2, ar3, art, &c.

For a ar: ar : ar2 :: ar2 : ar3, &c.

274. The constant multiplier, by which any term is derived from the preceding, is called the Common Ratio, and it may be found by dividing the second term by the first, or any other term by that which precedes it.

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

sion.

Let a, ar...ar"......a p2”...........a p3”.....&c. be the progression, then a, ar", ar2n, ar31, &c. are at the interval of n terms, and form a geometrical progression, whose common ratio is 7".

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

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

and r being thus known, all the terms of the progression ar, ar3, ar3, &c. are known.

277. To find the sum of a series of quantities in geometrical progression.

Let a be the first term,

the common ratio, n the number

of terms, and s the sum of the series:

Then s = a + ar + ap + + a pn−2+ a p2−1,

...

-1

ar+ar2+...+ apr−2+ apr-1+ ar",

Subtracting r8 − s = ar2 — a ;

rs

.. S=

COR. 1. If l be the last term, i = a rn-1, therefore

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

COR. 2.

value of ",

When

is a proper fraction, as n increases, the or of ar", decreases, and when n is increased without limit, a becomes less, with respect to a, than any magnitude that can be assigned. Hence the sum of the series,

This quantity

a

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 continued sine fine, for this series arises from the divi

Ex. 1.

To find the sum of 20 terms of the series, 1, 2, 4,

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

Ex. 3. Required the sum of 12 terms of the series

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

a

, yet,

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

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

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

Let 123123123 &c. = s; then, as in Art. 277, multiply both sides by 1000; and 123.123123123 &c.

=

1000s, and by subtracting the former equation from the latter, 123 = 9998;

[279. In a Geometrical series continued in inf. any term

is >, = or <, the sum of all that follow, according as the

1

...... be the series.

Then the sum of the series after n terms is the sum of

DEF.

HARMONICAL PROGRESSION.

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

280.

::

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

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

then by Def. 4: C: A-B B - C;

:

Again, B D :: B-C: C - D ;

therefore BC - BD = DB - DC,