BINOMIAL THEOREM.] A L G E B K A 555 Ex. It is required to find two square numbers whose sum is a given square number. Let a 2 be the given square number, and x 2 , y 2 , the num bers required ; then, by the question, x 2 + y 2 = a 2 , and y = /a 2 - x 1 . This equation is evidently of such a form as to be resolvable by the method employed in case 1. Accord ingly, by comparing / 2 - # 2 with the general expression 2 , we have g = a, 5 = 0, c= - 1, and substi tuting these values in the formulas, and also - n for + m, we find 2an a(n 2 -l) If a = n ij r; there results x = 2n, y = n 2 -!, a = n 2 + 1. Hence if r be an even number, the three sides of a rational If r be an 2 AA2 - 1,1- 1+1. , v odd number, they become (dividing by 2) r, , 2 2 For example, if r = 4; 4, 4 - 1, 4 + 1, or 4, 3, 5 are the sides of a right-angled triangle; if r 7, 7, 24, 25 are the sides of a right-angled triangle. SECT. XVI. THEOREMS OF EXPANSION. 1. Binomial Theorem. 122. To demonstrate this theorem, which has for its object the expansion of (a + x) n in the form P + Q^ + Ao; 2 + B# 3 + &c., we shall first find P and Q; and then de termine the other coefficients A, B, &c. in terms of P and Q. (1.) 1 + 5 a it being assumed that the power of a product is the pro duct of the powers of the factors, whatever be the index. (2.) Let n be a whole number. Since &c. ; if we assume (1 + x}*~ 1 = 1 + (n - 1) x + &c., and multiply both sides by 1 + x, we shall obtain (1 + x)* = 1 +nx + &c. ; whence our induction is complete to prove that the numerical of coefficient x is the same as the index, (3.) Let 11 be a positive fraction - or (1+ ar)!j= (1+ ar)" = 1 +px + &c. = 1 + +&c. +&c.) , + &c. (Case 2.) (4.) If n be negative = (CaseS.) x = 1 mx + &c. by division. Hence, generally the numerical coefficient of x is the same as the index. To obtain A, B, &c., in terms of the first and second terms, we break up x into two parts, y, z, which enables us to write the expression 1 + x in two different ways : 1st, retaining the parts of x in connection; 2d, disseveri.ig them. In tie first form we simply multiply out, and thus exhibit a result not dependent on the properties of an index, except in so far as relates to the first and second terms. In the second form we apply the properties of an index to every term. The consequence is, that the latter form, bearing a more intimate connection with the pro perty of an index than the former, is more determinate than the other. The comparison of the two results com pletes the demonstration. I. (1+aO" II.
- +B?/ 3 +&c.
+ nz + 2Ayz + 3B(/ 2 a + &c. + &c. + &c. 1 + 2 i + &C. 1 + ny + Ay 2 + &c. + nz + n(n- l)zy + A(n - + &C. + &C. +&C. zy 2 + &c. NOAV, as these two expansions are the expansions of the same thing in the same form, the coefficients of z, zy, zy 2 , &c., must be the same in both. Comparing them, we get n = n, 2 A = n(n - 1 ), 3B = A( - 2) &c. A = n(n 1) B = nQ-1) (ra-2)
- 1 . 2 . ff ~~
1.2 3 &c. = &c. ; and finally, whatever n be, (n l) n(?i 1) (n 2) ~ x* + -A JL^ _ 1 . ^ .Z.A Cor. 1. If re is a positive whole number, the series is finite, since every term after the (n+ l)th will involve n-n as a factor. Cor. 2. Since the coefficients, when the index is a whole number, are the results of simple multiplication, they are necessarily whole numbers, i.e., any such expression as n(ro-l)(re-2) . is a whole number when n is such 1 . 2i . o Cor. 3. The sum of the numerical coefficients is 2", for it is equal to (1 + l) n , as will appear if we write 1 for x. Cor. 4. The sum of the coefficients in the even places is equal to the sum of the coefficients in the odd. This will appear if we write - 1 in place of x. Cor. 5. If the index is a whole number, the coefficients from the end are the same as those from the beginning; for they occur at the beginning of (x + 1) B in the same positions as at the end of (1 + #)*. Cor. 6. The product 1 . 2 . 3 .... r is sometimes expressed by the abbreviated form r._ With this notation the coeffi- I n cient of x r in (1 + #)" may be written- Cor. 7. The sum of the squares of the coefficients of (1 + x)* is the coefficient of x* in the expansion of (1 + a;) 2 ", 2n and is equal to T^=CO 1 (In Ex. + &c. Examples. -2 -2 .-3 , -2. -3.-4 1 1, , 2 X + 1 .2.3 -2x+3x 2 -4
- 3 + &C.
1 _ nv -1 + 1) 9 -. V L Rrt> and = 1 + nx + + & c . Ex. 2. Find the coefficient of x 7 in (x + x 2 + tf f
The expression may be written
