299

CHAPTER XIII.

ON

PROPORTION AND PROGRESSION.

§ I. ARITHMETICAL PROPORTION AND PROGRESSION.,

403. ARITHMETICAL PROPORTION is the relation which two numbers, or quantities, of the same kind, have to two others, when the difference of the first pair is equal to that of the second.

404. Hence, three quantities are in arithmetical proportion, when the difference of the first and second is equal to the difference of the second and third. Thus, 2, 4, 6; and a, a+b, a+2b, are quantities in arithmetical proportion.

405. And four quantities are in arithmetical proportion, when the difference of the first and second is equal to the difference of the third and fourth. Thus 3, 7, 12, 16; and a, a+b, c, c+b, are quantities in arithmetical proportion.

406. ARITHMETICAL PROGRESSION is, when a series of numbers or quantities increase or decrease by the same common difference. Thus 1, 3, 5, 7, 9, &c. and a, a+d, a+2d, a+3d, &c. are an increasing series in arithmetical progression, the common differences of which are 2 and d. And 15, 12, 9, 6, &c. and a, a-d, a-2d, a-3d, &c. are decreasing series in arithmetical progression, the common differences of which are 3 and d.

407. It may be observed, that GARNIER, and other European writers on Algebra, at present, treat of arithmetical proportion and progression under the denomination of equi-differences, which they consider, as BONNYCASTLE justly observes, not without reason, as a more appropriate appellation than the former, as 'the term arithmetical conveys no idea of the nature of the subject to which it is applied.

408. They also represent the relations of these quantities under the form of an equation, instead of by points, as is usually done; so that if a, b, c, d, taken in the order in which they stand, be four quantities in arithmetical proportion, this

relation will be expressed by a-b-c-d; where it is evident that all the properties of this kind of proportion can be obtained by the mere transposition of the terms of the equation.

409. Thus, by transposition, a+d=b+c. From which it, appears, that the sum of the two extremes is equal to the sum of the two means: And if the third term in this case be the same as the second, or c=b, the equi-difference is said to be continued, and we have

a+d=2b; or b=(a+d) ; where it is evident, that the sum of the extremes is double the mean; or the mean equal to half the sum of the extremes.

410. In like manner, by transposing all the terms of the original equation, a-b-e-d, we shall have b-a=d—c ; which shows that the consequents b, d, can be put in the places of the antecedents a, c; or, conversely, a and c in the places of b and d.

411. Also, from the same equality a-b-c-d, there will arise, by adding m-n to each of its sides,

(a+m)—(b+n)=(c+m)−(d+n); where it appears that the proportion is not altered, by augmenting the antecedents a and c by the same quantity m, and the consequents b and d by another quantity n. In short, every operation by way of addition, subtraction, multiplication, and division, made upon each member of the equation, a-b-c-d, gives a new property of this kind of proportion, without changing its nature.

412. The same principles are also equally applicable to any continued set of equi-differences of the form a-b-bc=c-d=d-e, &c. which denote the relations of a series of terms in what has been usually called arithmetical progres

sion.

413. But these relations will be more commodiously shown, by taking a, b, c, d, &c. so that each of them shall be greater or less than that which precedes it by some quantity d'; in which case the terins of the series will become

a, a±d', a±2d', a±зd', a±4d', &c.

Where, if / be put for that term in the progression of which the rank is n, its value, according to the law here pointed out, will evidently be

l=a+ (n-1)d';

which expression is usually called the general term of the se

ries; because, if 1, 2, 3, 4, &c. be successively substituted for n, the results will give the rest of the terms.

Hence the last term of any arithmetical series is equal to the first term plus or minus, the product of the common difference, by the number of terms less one.

414. Also, if s be put equal to the sum of any number of terms of this progression, we shall have

s=a+(ad)+(a±2d')+

+[a±(n−1)d'].

...

And by reversing the order of the terms of the series, s=[a±(n−1)d']+[a±(n−2)d']+ . . . (a±d')+a. Whence, by adding the corresponding terms of these two equations together, there will arise

2s=[2a (n-1)d']+[2a+(n−1)d'], &c. to n terms.

And, consequently, as all the n terms of this series are equal to each other, we shall have

28=n[2a (n-1)d'], or s=[2a+(n-1)d'] .. (1).

415. Or, by substituting for the last term a(n-1)d', as found above, this expression (1) will become

s=(a+1)

(2).

Hence, the sum of any series of quantities in arithmetical progression is equal to the sum of the two extremes multiplied by half the number of terms.

It may be observed, that from equations (1) and (2), if any three of the five quantities, a, d', n, l, s, be given, the rest may be found.

416. Let l, as before, be the last term of an arithmetic series, whose first term is (a), common difference (d'), and number of terms

l-a (n); then la+ (n-1)d'; ... d'=N -1 ate terms between the first and the last is n

then n—1=m+1. Hence, d'

l-a m+I'

Now the intermedi

-2; let n—2=m,

which gives the fol

lowing rule for finding any number of arithmetic means between two numbers. Divide the difference of the two numbers by the given number of means increased by unity, and the quotient will be the common difference. Having the common difference, the means themselves will be known.

Example 1. Find the sum of the series 1, 3, 5, 7, 9, 11, continued to 120 terms.

Here a=1,

d'=2,

n=120,

&c.

s=[2a+(n−1)d']}=12o[2×1+(120–

1)2]=14400.

Ex. 2. The sum of an arithmetic series is 567, the first term 7, and the common difference 2. What are the number of terms? Here s=567, ... 2s=n[2a+(n−1)d]=n[14+(n−1)2] =14n+2n2--2n=1134; .. n2+6n+9=

1576

a=7, d'=2; 576, and n=21. Ex. 3. The sum of an arithmetic term 5, and the number of terms 30. ference?

series is 1455, the first What is the common dif

Ans. 3.

series is 1240, common dif

Ex. 4. The sum of an arithmetic ference 4, and number of terms 20. What is the first term?

Ans. 100.

Ex. 5. Find the sum of 36 terms of the series, 40, 38, 36, 34, &c.

Ans. 108.

Ex. 6. The sum of an arithmetic series is 440, first term 3, and common difference 2. What are the number of terms?

Ans: 20.

Ex. 7. A person bought 47 sheep, and gave 1 shilling for the first sheep, 3 for the second, 5 for the third, and so on. What did all the sheep cost him? Ans. 110/. 9s.

Ex. 8. Find six arithmetic means between 1 and 43.

Ans. 7, 13, 19, 25, 31, 37.

§ II. GEOMETRICAL PROPORTION And progression. 417. GEOMETRICAL PROPORTION, is the relation which two numbers, or quantities, of the same kind, have to two others, when the antecedents or leading terms of each pair, are the same parts of their consequents, or the consequents of their antecedents.

418. And if two quantities only are to be compared together, the part, or parts, which the antecedent is of the consequent, or the consequent of the antecedent, is called the ratio; observing, in both cases, to follow the same method.

419. Direct proportion, is when the same relation subsists between the first of four quantities, and the second, as between the third and fourth.

Thus, a, ar, b, br, as in direct proportion.

420. Inverse, or reciprocal proportion, is when the first and second of four quantities are directly proportional to the reciprocals of the third and fourth.

Thus, a, ar, br, b, are inversely proportional; because, a, ar, 1 1 are directly proportional.

br' b'

421. The same reason that induced the writers mentioned

in (Art. 407), to give the name of equi-differences to arithmetical proportionals, also led them to apply that of equi-quotients to geometrical.proportionals, and to express their relations in a similar way by means of equations.

Thus, if there be taken any four proportionals, a, b, c, d, which it has been usual to express by means of points, as below,

a:b::c: d.

This relation, according to the method above-mentioned,

α C

will be denoted by the equation (Art. 24); where the equal ratios are represented by fractions, the numerators of which are the antecedents, and the denominators the consequents. Hence, ad=bc.

422. And if the third term c, in this case, be the same as the second, or c=b, the proportion is said to be continued, and we have ad=b2, or b=ad; where it is evident, that the product of the extremes of three proportionals, is equal to the square of the mean: or, that the mean is equal to the square root of the product of the two extremes.

423. Also, from the equality,

for, by adding or subtracting 1 from each side of the

a

abcd

and a±b:b::

equation; then +1=1;.."

c±d: d.

Hence, when four quantities are proportionals, the sum or difference of the first and second is to the second as the sum or difference of the third and fourth, is to the fourth.

424. In like manner, if a: b::c: d; then, ma: mb::c:

Hence, when four quantities are proportionals, if the first and second be multiplied, or divided by any quantity, and also the second and fourth, the resulting quantities will still be proportionals.

a

an

=

and

425. Also, if a: b::c: d; then;

an: bn :: c2: dr; where n may be any number either integral

or fractional.