An algebra problem by Puneet Pinku

Algebra Level 3

2 x 99 + 3 x 98 + 2 x 97 + 3 x 96 + . . . . . . . + 2 x + 3 = 0 2x^{99}+3x^{98}+2x^{97}+3x^{96}+.......+2x+3 = 0 . Find the sum of the real roots of the above equation

− 4 -4 − 3 2 -\frac 32 None of the other choices 3 i 3i − 1 2 -\frac 12 − 1 3 -\frac 13 − 6 7 -\frac 67

Puneet Pinku by Puneet Pinku , 20, India

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1 solution

Apply the geometric sum to see that the entire expression is. (3+2x)(x^50 -1)/(x^2-1). So one of the real solution is -3/2. The solutions of x^50 = 1. Is the 50th roots of unity. Which are of the form e^(2ikpi/50). Where k is an integer from {0,1,2.....49}. Now we see that only for k=0 and k= 45, the 50th roots of unity are real. But for those values...i.e... 1 and -1 the expression of x^2-1 which is on the botton becomes 0 and hence the total expression becomes undefined. So the only real solution to the expression is -3/2

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