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is easily found; thus, if a = 98, and b = 99225, we get y =

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25. Having a, the sum, and b, the product of the extremes of three numbers or quantities, in harmonical progression, to find the progression.

Ans. The mean term equals

a + √ a2

2

8b

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equals the sum of the extreme terms; conse

quently, the progression is easily found; thus, if a

=

143,

and b 2205, we get 45 for the mean, and 35 and 63 for the extremes.

26. "What number is that which, when divided by the product of its two digits, the quotient is 3, and if 18 be added to it, the digits will be inverted?"

will

If y is the unit's, and a the ten's digit, then 10x + y express the number, and = 3 by the first condition;

10x + y

xy

also, 10x+y+18=10y+x by the second condition.

From the elimination of y we get the quadratic æ2.

5 2 x=

3 3' whose solution gives x=2, and 24 will be the sought number.

27. There is a number consisting of three digits, such that the middle digit is a geometrical mean between the other two, and that the sum of the extreme digits, when diminished by 1, equals twice the middle digit. Moreover, supposing the sum of the digits to equal 19, and that the hundredth's digit is greater than the unit's digit, it is proposed to find the number. Ans. 964.

28. If a body moves 1 yard the first minute, 3 yards the second minute, 5 yards the third, 7 yards the fourth, and so on, then how long will the body be in moving 400 yards? Ans. 20 minutes.

29. Having the product of the sum of two numbers by the greater equal to 153, and the product of the sum by the less equal to 136, to find the numbers. Ans. 8 and 9.

30. The product of two numbers is 28, and after the first is increased by 6 and the second diminished by 4, the product of the resulting numbers is 30; then what are the numbers? Ans. 4 and 7.

31. Supposing the sum of a certain principal and its amount at compound interest for two years to be $2102.50, and that the amount of the same principal, at the same compound rate, in four years, is $1215.50625; then what is the principal? and the rate per cent. per annum?

Ans. The principal is $1000, and the rate 5 per cent.

32. The sum of the squares of two numbers equals 61, and their sum is to their product as 11 to 30; then what are the numbers? Ans. 6 and 5.

33. The sum of the reciprocals of two numbers is 5, and the sum of the squares of their reciprocals is 13; then what are the numbers? Ans.

1

1

2

and g

3

34. Having the sum of the nth powers of a number or quantity, and of its reciprocal, equal to a, to find it.

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35. Having the sum of the n" powers of two numbers or quantities equal to a, and the sum of the n" powers of their reciprocals equal to b, to find them.

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36. Having the sum of the squares of two numbers or quantities equal to a, and the sum of the reciprocals of the numbers or quantities equal to b, to find them.

If x and y stand for the numbers or quantities, the con1 1 x + y ditions give the equations a+ya and + =

Y xy

= -b.

2

From the second of these equations we get 2ay-Z (x+y)

=

0, which, added to the first equation, reduces it to the quadratic (x + y)3 — {(x + y) = a, whose solution (regarding

2

1+ √ab2 + 1

x and y as positive) gives x + y = b

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and xy are known, it is easy to find the values of x and y.

SECTION XV.

METHOD OF UNDETERMINED COEFFICIENTS.

(1.) IF an equation of the form A + B + Cx2 + etc., = A' + B'x + C'a2 + etc., (1), in which A, B, C, etc., A', B', C', etc., do not contain a, must be satisfied by any value which may be given to æ, then shall A = A', B = B', C=C', and so on, or (regarding A and A' as coefficients of ao1) the coefficient of any power of x in one member of the equation equals the coefficient of the same power of x in the other member.

For, by transposition, etc., (1) gives

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which clearly can not be satisfied for all the values that may be given to a (or so as to leave a arbitrary), except by putting the numerator and denominator of the right member separately equal to 0; consequently, we must have A = A', and B - B' + (C — C' ́)x + (D − D′)x2 + etc. = 0, (3).

UNDETERMINED COEFFICIENTS.

It may in like manner be shown from (3) that we must have B = B', and then that C = C', and so on, as required. Hence, if for A', B', C', etc., we put their equals, A, B, C, etc., (1) will become the identical equation A+ B+ Cx2+ etc. A + Bx + Cx2+ etc.

=

Universally, if we have the equation A + Bø(x) + C¤′(x) + etc. A' + B'p(x) + C'p'(x) + etc., (4), such that A, B, C, etc., A', B', C', etc., are independent of any values which p''(x) p'''(x), etc., and so p(x), p′(x), p′′(x) etc.,

'(x) o'x p(x) px

etc., "

'(x)' '(x)

=

on (which are functions of x, or dependent on it), can have, then it is clear that, by a like reasoning to that used above, we can show that. we must have A A', B = B', C = C', and so on; consequently, by the substitution of the values of A', B', C', etc., (4) becomes the identical equation A + Bp(x) + C¤'(x) + etc. = A + B¤(x) + Cp'(x) + etc., in which a is clearly arbitrary.

(2.) Since any function of x, when developed into a series of ascending or descending powers of, or of any other functions of a (which the nature of the case may require), ought to be considered and treated as being identical with the series which represents it, it follows (from what has been done) that, after we have determined a suitable form for the development, we may represent the coefficients of its terms. by any undetermined letters, as A, B, C, etc., and then determine the values of the letters by equating them to the coefficients of the corresponding powers (or functions) of in the given function, so that the given function and its development shall constitute an identical equation; and it is in this process, that the Method of Undetermined Coefficients essentially consists.

(3.)

APPLICATIONS OF THE METHOD TO EXAMPLES.

1. To convert the fraction

a into a series, arranged b + cx

according to the ascending powers of x.

If we divide the numerator and denominator of the frac

α

=

tion by b, and put % = a' and b', the fraction will become

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a'

1+ b'x'

assume

; and it is clear (from actual division) that we may

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a' + Ax + Bx2 + Ca3 + etc., for the de

velopment of the fraction.

Freeing the equation from the fraction, we get a' = a' + (A + a'b')x + (B + Ab')x2 + (C + Bb′),3 + etc., which must be an identical equation.

Consequently, if we equate the terms which do not contain , and put the coefficients of x, x2, x3, etc., equal to 0 (since the coefficients of a, a, a, etc., in the first member of the equation equal 0), we get the identical equation a' a', and the equations A + a'b' = 0, B + Ab′ = 0, C + Bb' = 0, etc., or A = a' × (b'), B = A × (→ b'), C= B × (− b'), etc.

Hence, we have

a

=

a' = a' + a' × (— b'x) + b + cx 1 + b'x

a' × (−b'x) × (—b'x) + a' × (− b'x) × (— b'x) × (− b'œ) + α acx ac2x2 ac3x

etc. = 7

b

+ b2

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+ etc.; which is such that if

cx

we multiply any one of its terms by the product gives

the next successive term.

Remark.-If we put x =

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; e + by consequently, developing c + by

into a series arranged according to the ascending powers of y,

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Ans. 1+ 3x + 9x2 + 27x+81x + etc.

into a series, arranged

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