429. To find the Sum of the whole Progreffion, (called the total Sum) having the two Extremes and Number of Terms given; the Rule is: The Sum of the two extreme Terms, being multiplied by the Number of Terms, will be equal to twice the Sum of the whole Progreffion*.

430. Example. Suppofe there are io Pieces of Cloth, the first containing 2 Yards, the fecond 4 Yards, the third 6 Yards, and fo increafing by the continual Addition of 2: How many Yards are there in all the Pieces?

Solution. The Number of Terms io, — i — 9, † 427. and 9 x 2 = 18, + 2 = 20† the last Number; hence, by the above Rule, 20 + 2 = 22, x 10 = 220 Yards twice the Sum of the whole Progreffion; 2202110 Yards (or 22 x 5, half the Number of Terms) = the Sum of the whole Progréffion, which was required.

and

CHAP. XXXIV.

Of GEOMETRICAL PROGRESSION, or GEO-
METRICAL PROGRESSION continued.

431. W increafes, or decreases, by the continual

HEN a Rank, or Series of Numbers,

Multiplication or Division of one common Number, fuch

*Any Series may be expreffed by the continual Addition of the common Difference to the leaft Term, or the continual Subtraction of the fame from the greatest Term; thus a being =leaft Term, d common Difference, n = Number of Terms, g = greatest Term, s = Sum of all the Terms, we may exprefs the Series two Ways, a a+d.a+2d. a + 3d. &c. = s. viz, the Sum of 2 d. g 30 d. &c. = s. And therefore, if we add thefe two Series together, their Sum muft be 2s. But it is evident that the Sum of any two correfponding Terms is constantly the fame, viz. a+g, and the Sum of all the Terms in both ag taken n Times; that is, a + g X = 28. 2 E.D.

Sg.g-d.g.

fuch Rank, or Series, is called a Geometrical Progreffion. As, for Instance, 2. 4.8. 16. 32 . &c. is a Rank or Series in Geometrical Progreffion increafing, the common Multiplier being 2. This Series may

be expreffed in a decreasing Order thus, 32. 16. 8. 4. 2, and here the common Divifor is 2. Now this common Multiplier, or Divifor, is called the Geometric Ratio.

432. The Ratio multiplied by the Ratio is called the fecond Power of the Ratio; and the Ratio multiplied by the Ratio, and that Product by the Ratio, is called the third Power of the Ratio; and the third Power of the Ratio, multiplied by the Ratio, is called the fourth Power of the Ratio, &c. &c.

433. Any Term, in an increasing Geometrical Progreffion, is equal to the first Term multiplied by that Power of the Ratio which is denoted by its Distance from the first Term*. (Whence if over the Terms of the Geometrical Progreffion we put an Arithmetical Progreffion whofe firft Term is o, and common Difference 1, the Terms of the Arithmetical Series (called the Indices) will denote the Power of the Ratio, in any of the correfponding Terms, of the Geometric Progreffion.)

434. Example.

A Gentleman, as he did ride

Near to a pleasant Common-Side,
Ten Shepherdeffes chanc'd to meet,
Driving their Flocks, whom he did greet,
God fpeed you well; and may you be
As happy as you're fair (faid he :)
Profper your Flocks, and may they thrive;
Tell me how many Sheep you drive?
One of the Damfels ftraight reply'd,
Sir, you fhall foon be fatisfy'd:

P

For,

* This will plainly appear, by only reprefenting a Geometric Progreffion by Letters, viz. putting a leaft Term, r common Multiplier, the Indices are o. 1. 2. 3. 4. &c.

And the Geometric Progreffion is a. ar.

arr. arrr. arrrr, S.

For, if for one of us you

do

Count one Sheep, for the next count two,
For the third four, for the fourth eight,
So doubling at each Maid aright,
At the laft Maid the Sum will be
As many as the Sheep you fee.

Quere the Number of Sheep? Solution. Here, the Distance of the last Term from the first being 101 9, we have, by Art. 433, the last Term = 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 1 512 the Number of Sheep which was required.

=

=

435. But fince this Method of finding the Power of the Ratio, if the Number of Terms be many, is very troublesome, it is many Times convenient to make Ufe of the following Contrivance, viz. It plainly follows from the latter Part of Article 433, that the Product of the Powers of the Ratio under any two Indices will be equal to the Power of the Ratio correfponding to that Index, which is equal to the Sum of the two Indices; hence the above Question may be folved, by fetting down some of the Powers of the Ratio with their correfponding Indices, s Indices

as,

I. 2

3

4.5

&c. Powers of the Ratio 2.4.8.16.32. &t. Here, the Distance of the first and last Term being 9 by the Question, we are to take two fuch Indices as being added together fhall be equal to 9, viz. 5 + 49; now under 4 is 16, and under 5 is 32, . 16 × 32=512 = the 9th Power of the Ratio, which, being multiplied by the firft Term of the Series (here 1) gives 512 x 1=512 Sheep as before.

436. It may also be folved by the Addition of 3 or more Indices, as, for Example, by the Indices, 4, 3, and 2, for 4 +3 +2 = 9; and under 4 is 16, under 3 is 8, and under 2 is 4. And 16 x 8 x 4 = 512 the 9th Power of the Ratio as before.

437. But, perhaps, the moft compendious Method is, not to write down any Terms of the Series at all, but to imagine the Indices fixed over the Terms in your Mind, and work as follows, viz. The In

dex + (itfelf) 1 = 2,. the Power of the Ratio under that Term whofe Index is 2, is 2 x 2 = 4 the Ratio x by the Ratio; and Index 2 + (itself) 2 ≈ 4, *.*, the Power of the Ratio belonging to the third Term, viz. 4, being multiplied by itfelf, that is, 4×4=16= the Power of the Ratio belonging to the fifth Term; but the Index 4+ (itfelf) 4 =8 the Index belonging to the 9th Term, . 16 x 16 256 the Power of the Ratio belonging to the ninth Term; laftly, the Indices 8+ 19 the Index of the ioth Term, . 256 x 2 = 512= the Power of the Ratio belonging to the tenth or laft Term as before.

438. In any Geometrical Progreffion, as any Antecedent is to its Confequent, fo is any other Antecedent to its refpective Confequent. This is evident, for by the Nature of fuch Progreffions, if r denote the common Multiplier, any Confequent is r Times

its Antecedent.

439. Therefore, in a continued Proportion, all the Terms, except the laft, are called Antecedents, and all the Terms, except the firft, Confequents.

440. Whence it follows *, that as the leaft (or any other) Term is to its Confequent (the Term next following) fo is the Sum of all the Antecedents of the whole Progreffion to the Sum of allthe Confequents.

441. And from hence is deduced the following Rule, to find the Sum of all the Terms, the two extreme Terms and common Multiplier being P 2 given,

* Let a. ar. arr. arrr. arrrr. arrrrr, &c. be the Series; A all the Antecedents, Call the Confequents, then, atar+arr tarrr +arrrr, &c. A; and ar+arr+arrr+arrrr+arrrrr, &c. = C; Hence it is evident by a bare Infpection, that the Sum of all the Confequents is r Times the Sum of all the Antecedents; but each particular Confequent is alfo r Times its refpective Antecedent, and, confequently, any Antecedent, its Confequent, the Sum of all the Antecedents, and Sum of all the Confequents, are four Quantities in direct Proportion, viz. as any Antecedent: its Confequent :: A C. 2.E.D.

given, viz. Multiply the greatest Term by the common Multiplier, from which Product fubtract the firft Term, and divide the Remainder by one lefs than the common Multiplier, and the Quotient will be the Answer*.

442. Example, or Question 2. Suppofe A agrees with with B to fell him a Houfe, which has 12 Windows, if he will put 2 Pence in the first Window, 6 Pence in the Second, 18 Pence in the Third, and so on, multiplying by three each Time, thro' all the Windows: What would the House cost?

Solution. Here, the Number of Terms being 12, we are first to raise 3, the Ratio, to the eleventh Power; to do which, the fhorteft Method is, 3 x 3

=

9 the fecond Power, and 9 x 981 = the fourth Power, then 81 x 81 = 6561 the eighth Power; and, 8+2 being 10, 6561 x 959049 the tenth Power, and, 10 + 1 being 11, we fhall have 59049 × 3 = 177147 the eleventh Power, ... 177147 x 2 = 354294 the twelfth or greatest Term; now, to find the Sum of all the Terms, we have, first, 354294 × 3 = 1062882, from which fubtracting the firft Number 2, we get 1062880 for a Dividend, which, divided by 312, gives 531440 Pence 2214l. 6s. 8 d. for the Answer.

443. The Reafons for our having been fo fhort on Arithmetical, and Geometrical Progreffion are, first, that they are of no Ufe in common Arithmetic; secondly, that we shall have Occasion to discourse more largely of them, when we treat of Algebra: And,

*

for

a:

* Let s the Sum of the Series, a the least Term, d the common Multiplier, then da the next Term to the leaft; also put g the greatest Term, then s— g: the Sum of all the Antecedents, and s-a= the Sum of all the Confequents; hence, as 440 da :: s-g: s—a. •: da s· ・gda=† sa—aa, ., dividing by a, +185 we have ds-gd=1s-a, hence, by adding gd to each Side of the Equation, we shall get ds = || gd+sa, and, by fubtracting s § 36. from this, ds. -S= §gd-a; but d's s = d. Ixs, d ¶ 108. xs=gd-a; which, divided by d1, gives s=¶

108.

|| 22.

gd. -a

2.E. D.