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Mean the roots (Posted on 2019-03-13) Difficulty: 3 of 5
What is the harmonic mean of the roots of the following polynomial?

2019x2018-2018x2017+2017x2016-2016x2015+...+3x2-2x+1 = 0

No Solution Yet Submitted by Danish Ahmed Khan    
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Solution Solution Comment 1 of 1
First divide all the coefficients of the polynomial by 2019: x*2018 - (2018/2019)x^2017 + (2017/2019)x^2016 - ... + (3/2019)x^2 - (2/2019)x + 1/2019 = 0.

The formula for the harmonic mean is H = 2018 / (1/x_1 + 1/x_2 + ... + 1/x_2018).

Multiply the numerator and denominator by the product of all 2018 terms:
2018 * (x_1*x_2*...*x_2017*x_2018) / (x_2*...*x_2017*x_2018 + x_1*x_3*...*x_2017*x_2018 + x_1*x_2*x_4*...*x_2017*x_2018 + ...... +  x_1*x_2*...*x_2016*x_2018 + x_1*x_2*...*x_2016*x_2017)

Now the numerator is 2018 times the constant term of the polynomial and the denominator is the linear coefficient of the polynomial.  Then H = 2018 * (1/2019) / (-2/2019) = -1009

Edited on March 15, 2019, 1:31 am
  Posted by Brian Smith on 2019-03-15 01:29:48

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