By C. W. Celia, A. T. F. Nice, K. F. Elliott

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For the required equation sum of the roots = tX 2 + fJ 2 + y2 = (IX+ fJ + y)2 - 2tXfJ- 2fJy- 2ytX = p2-2q product of the roots= tX 2fJ 2y2 = (tXfJy) 2 = r2 (X2p2 + p2y2 + }'21%2 = (tXfJ + fJy + ')11X)2- 2tX2{3y- 2tXfJ2y- 2tX/J}'2 = (1%{3 + fJy + }'1X) 2 - 21%/J}'(IX + {3 + ')') = q2 -2(-r)(-p) = q 2 -2pr 21 Advanced mathematics 1 :. the required equation is x 3 - (p 2 - 2q)x 2 + (q 2 - 2pr)x- r 2 = 0. 2 1 Find the sum and product of the roots of the following equations. (a) 2x 2 +3x-4=0 (d) 5x 2 +2x-3=0 (b) 3x 2 - x- 6 =0 (e) x 2 +3 = 1 X (c) 9x 2 +24x+ 16 = 0 1 (f) 1 +- = x X In each case state the nature of the roots.

The definition of a sequence requires (a) the first term, (b) the number of terms, (c) the law by which successive terms can be found. , ... , un, the first term is u 1 , the sequence has n terms and the law relating the terms will be given by the general term u,. Consider the sequence often terms 1, 3, 5, 7, 9, 11, 13, 15, 17, 19. In this sequence each term is 2 more than the preceding term. and u 1 = 1, u2 = 1 + 2 = 3, u3 = 1 + 2(2) = 5, u4 = 1 + 3(2) = 7, u, = 1 + (r -1) (2) = 2r -1. When the difference between consecutive numbers in a sequence is constant (as it is in the example just considered), an expression for the general term u, can be obtained in the form ar + b.

2n-l · · ·(1) and 1 +2+4+8 + ... +2n-l ... e. series of the form a +ar+ar2 + ... +ar"-1, where a is the first term and r is the common ratio of successive terms. In series Algebra 2 && (1), a= 1 and r = t and in series (2) a= 1 and r = 2. Let Sn=a+ar+ar 2 + ... +arn- 1 Multiplying by r, rSn = ar + ar 2 + ... + arn- 1 + arn Subtracting, (1- r)Sn =a - ar" a(1-,n) giving Sn= 1 , (r ¥- 1). -r If I, m and n are three consecutive terms of a geometric progression, then m is said to be a geometric mean of I and n.