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Algebra Level 3

The polynomial x 2 + 2 x − 1 x^2+2x-1 has roots as α α and β β , let us define S n S_n as α n + β n α^n+β^n , then which of the following statements are true (can be done without a calculator or help of a program)

Do not try to calculate S 7 with a pen and paper \textbf{Do not try to calculate } S_7 \textbf{ with a pen and paper}

This problem is not original \tiny\text{This problem is not original}

S 7 + 2 S 6 + S 5 = 0 S_7+2S_6+S_5=0 S 7 − 2 S 6 = S 5 S_7-2S_6=S_5 S 5 + 2 S 6 = S 7 S_5+2S_6=S_7 S 7 + 2 S 6 = S 5 S_7+2S_6=S_5 S 7 + S 5 = 2 S 6 S_7+S_5=2S_6

Jason Gomez by Jason Gomez , 16, India

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1 solution

Chew-Seong Cheong
Feb 20, 2021

By Newton's sums or Newton's identities

S 7 = ( α + β ) S 6 − α β S 5 By Vieta’s formula: α + β = − 2 , α β = − 1 S 7 = − 2 S 6 + ( 1 ) S 5 ⟹ S 7 + 2 S 6 = S 5 \begin{aligned} S_7 & = \blue{(\alpha + \beta)} S_6 - \blue{\alpha \beta} S_5 & \small \blue{\text{By Vieta's formula: }\alpha + \beta = -2, \alpha \beta = -1} \\ S_7 & = \blue{-2} S_6 + \blue{(1)}S_5 \\ \implies S_7 + 2S_6 & = S_5 \end{aligned}

Reference: Vieta's formula

2 Helpful 0 Interesting 2 Brilliant 0 Confused

Thank you sir, for posting a solution here

Jason Gomez - 3 months, 3 weeks ago

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