ALGEBRA [FUNDAMENTAL The sum "of their squares when increased by 1 is Now, either n or n 4- 1 is even, . . n(n 4-1)4-1 is odd ; hence the sum under consideration is 12 times an odd uuniber, whence the proposition. Additional Exanyjles in Symmetry, cfrc. Ex. l.(a + b + c) 2 + (a + b- c) 2 + (a + c - b) 2 + (b + c - a) 2 This is written down at once, from observing that a 2 occurs in each of the four expressions, and that 2ab occurs with a 4- sign in two, and with a - sign in the other two. There is no other form. Ex. 2.(a. + b + c) s + (a + b-c) 5 + (a + c-b) 3 + (b + c- a) 3 c= 2(a 3 4- b 3 + c 3 ) + 6(a 2 b + a 2 c + b 2 a + b 2 c + c 2 a + c 2 b) - I2abc. 1st, a 3 occurs 4- in three, and in one term. 2d, 3a 2 b occurs 4- in three, and in one term. 3d, When a, b, c are all units, the number resulting is 30; . . there are 30 terms, and as (1st) and (2d) make up 42, there fall to be subtracted 12, i.e., the coefficient of abc is - 12. Ex. 3. (ax + by + cz) 2 + (ax + cy + bz)- + (bx + ay 4- cz) 2 + (bx + cy + azf 4- (ex + ay 4- bz) 2 4- (ex + by + az) 2 = 2(a 2 + b 2 + c 2 ) (x 2 4- y 2 + z 2 ) 4- 4:(ab 4- ac + be) (xy + xz + yz). Ex. 4. The difference of the squares of two consecutive numbers is equal to the sum of the numbers. Ex. 5. The sum of the cubes of three consecutive num bers is divisible by the sum of the numbers. Ex. 6. If x is an odd number, x 5 - x is divisible by 24, and (z 2 4- 3) (x 2 + 7) by 32. Ex. 7. If (pq - r) 2 + 4(p 2 - q)(pr - q) 2 = 0, then will and 4(<7 2 -prf = (2gp - Zpqr + r 2 ) 2 . Ex. 8. Given x + y + z = Q, X 4- Y 4- Z = 0, to prove that (x 2 + XV + (f + Y2 )^ + (* 2 + Z2 )^ - (^ + X2 ) YZ + Let the left hand side equal the right + u ; then multi plying out, xyz(x + y + z) + J 2 i/z + ~Y 2 zx + Z 2 xy = XYZ(X + Y + Z) + * 2 YZ + y 2 ZX + z 2 XY 4- u, i.e., X?yz + Y*zx + (X. + Y)*xy = x- YZ 4- /ZX + (x + y) 2 XY 4- u , or, Wy(z + x) + Y*x(z + y) = z 2 Y(Z4-X)4-/X(Z4-Y)4-, or, - X 2 y 2 - Y-V = - z 2 Y 2 - y 2 X 2 4- u Ex. 9. If 4a 2 6 2 c 2 (z 2 + y 2 + z") (a^x 1 + b*f + c 2 * 2 ) = {(Z> 2 4- c 2 )alc 2 + (c 2 + a 2 )% 2 4- (a 2 4- & 2 > 2 *)} 2 , when a is greater than &, and b greater than c; then is y = 0. As the argument concerns y, multiply out, and arrange in order of powers of y. After reduction this results in (a 2 - c 2 )6 Y 4- 2 { (a 2 - c 2 )(6 2 - c> 2 x 2 4- (a 2 - c 2 )(a 3 - & 2 )c% 2 } &y + |(6 2 - c 2 )a 2 a; 2 - (a 2 - 6 2 )cV} 2 - 0. Now each, of these three terms is a 2^sitive quantity, if it be not zero, and as the sum of three positive quantities cannot be equal to zero, it follows that each term must be separately equal to zero, i.e., y = 0, and (b" c 2 )a 2 2 2 = (a 2 & 2 )c~2 2 . 17. Inequalities. The demonstrations of inequalities are of so simple and instructive a character, that a somewhat lengthened exhibi tion of them forms a valuable introduction to the higher processes of the science. In all that follows under this head, the symbols x, y, z stand for positive numbers or fractions, usually unequal. Ex. 1. x" + y* > 2.r?/. Because (x - y) 2 is 4- , whether x be greater or leas than y, it follows that x 2 -2xy + y 2 is +, i.e., is some positive number or fraction, It will be remarked that whun x and y are equal, the in equality rises into an equality, and this is common to all inequalities of the character under discussion. X 17 Cor. - + ->2 : i.e.. the sum of a fraction and its recipro y ^ cal is greater than 2. Ex. 2. x * + y 2 + z->xy + xz + yz. For x z + y 2 >2xi/, x 2 + z->2xz, if + z- >1yz ; which being added and divided by 2, gives the result required. Ex. 3. x m+n + y n+n >x m y n + :>:"// m . For (x m y m ) (x n y n ) is 4- , whether x be greater or less than y. As a particular case x 3 + ?/ 3 >x 2 y + xy^. Ex. 4. x> n +l>x 2n -* r + x~ r . For (x- H ~ 2r - 1) (x* r - 1) is positive. Cor. Cor. 2. Similarly, a; n 4-->.v n ~ 1 4--,,_i , C JJ 1 . i.e., as n increases x +- n increases, . . as a particular case > i X Ex. 5. If a, b, c are the sides of a triangle, a" + l>~ + r 2 >ab + ac + bc< 2(ab 4- ac + be). The former inequality is proved in example 2. For the latter we have (Euclid, I. 20), a<b + c .: a 2 <ab + ac. Similarly, b 2 <ab + be, c-< ac + be . .: a 2 + b 2 + c 2 < 2 (a!> + ac 4- be). Ex. 6. The arithmetic mean of any number of quantities (all positive) is greater than the geometric. (The arithmetic mean is the sum of the quantities divided by their number ; the geometric is that root of their product which is represented by their number.) Let the quantities be denoted by x 1} a*,, x 3 , . . . x n , the num bers 1, 2, 3, placed under the x, indicating order only, so that x may be read the first x, x 2 the second x, etc. Ex ample 1 gives - 1 -^- 2 > *JziX->, if we suppose the x and y of that example to be Jx lt *Jx., of the present. It also gives - 1 - 2 + - 4 ,- _ ia " ~~ v _ >/_,Jz _ In the same way we prove the proposition for 8, 1C, or any number of quantities which is a power of 2. For any other number, such, for instance, as 5, the following process is employed : The number is made up to 8 by the insertion of three quantities, each equal to the arithmetic mean of the other five, viz.,
Call this quantity y; then
