4: The population of a town increases annually in geometrical progression, and in four years is raised from 10000 to 14641. By what part of itself is it increased each year? Ans. To 5. Find 4 numbers in geometrical progression, such that the sum of the means shall be 36, and the sum of the extremes shall be 84. Ans. 3, 9, 27, and 81. 6. Insert 3 geometrical means between and §. Ans. 1,, . INDETERMINATE COEFFICIENTS. 174. An identical equation is one which is true for all possible values of the unknown quantities which enter it. Thus, 175. Every identical equation containing but one unknown quantity, can be reduced to the form of, p + qx + rx2 + &c. = p'+ q'x + r'x2+ &c. (1.) Or, by transposition, to the form, (p −p') + ( − q′)x + ( r — r')x2 + &c. = 0. (2.) 176. From the definition above given, the unknown quantity may have an infinite number of values, that is, it is indeterminate (Art. 74). 177. The coefficients of the different powers of the unknown quantity in Equation (2) being coefficients of indeterminate quantities, are called INDETERMINATE CO EFFICIENTS. 178. We have, from Equation (2), by transposition, (1 − q′)x+(r — p1);x2 + &c. = p'-p From the definition of an identical equation (Art. 174), it must be of such a form as to be satisfied for every possible value that may be attributed to x. Hence, Equations (1), (2), and (3), must be true for every value of ; consequently, for the value of 0. Substituting this value in the first member of Equation (3), reduces it to zero: hence, p'- p = 0, or p = p'. Making pp', in Equation (3), and dividing through by x, we have, q-q' + (r− r)x + &c. = 0. In the same manner as before, it may be shown that q = q', and by continuing the same course of reasoning, we can show that r = ', and so on; that is, The coefficients of the different powers of the un. known quantity, in Equation (2), are separately equal to 0; or the coefficients of the like powers of the unknown quantity in the two members of Equation (1) are separately equal to each other. This principle, called the principle of indeterminate coefficients, may be enunciated as follows: In any identical equation, containing but one inde terminate quantity, the coefficients of the like powers of this quantity in the two members, are separately equal to each other. 179. If an identical equation contains more than one indeterminate quantity, it may be shown by a similar course of reasoning, that the coefficients of the like powers and combinations of powers of these quantities, in the two members, are separately equal to each other. Hence, the following general principle: In any identical equation containing any number of indeterminate quantities, the coefficients of the like powers and combinations of powers of these quantities, in the two members, are separately equal to each other. The principle of indeterminate coefficients is much used in developing expressions into series. To illustrate, let it be required to develop the expression, 2 + 3x 3+4x+5x2 p + qx + ræ2 + sã3 + &c. (1.) Clearing of fractions, we have, 2 + 3x = 3p + 4px + 5p 202 + 5q23+ &c. From the principle of indeterminate coefficients, we have, Substituting these values in (1), we have the required In like manner, any similar expression may be developed into a series. Hence, the following RULE. Assume the expression equal to a series of the form, p + qx + rx2 + &c., in which p, q, r, &c., are quanti ties to be determined. Clear the equation of fractions, and place the coefficients of the like powers of the unknown quantity in the two members separately equal to each other. Then find, from the resulting equations, the values of p, q, r, &c., and substitute these values in the assumed development. |