Some kind of quadratic!

Algebra Level 5

a n + b n = 181521 a^n+b^n=181521

Let a a and b b be the roots of x 2 + 3 x + 27 = 0 x^2+3x+27=0 .

Find the smallest positive integral value of n n which satisfy the above equation.


The answer is 7.

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Mark Vincent Esmeralda Mamigo by Mark Vincent Esmeralda M… , 24

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2 solutions

Prashant Kr
Jan 25, 2016

using newton's summation we define

s n = a n + b n s_{n} = a^{n} +b^{n} so S 1 = − 3 S_{1}= -3 S 2 + 3 S 1 + 54 = 0 ⟹ S 2 = − 45 S_{2} +3S_{1} +54=0 \implies S_{2}= -45 S 3 + 3 S 2 + 27 S 1 = 0 ⟹ S 3 = 216 S_{3} +3S_{2} +27S_{1}=0 \implies S_{3}=216 S 4 + 3 S 3 + 27 S 2 = 0 ⟹ S 4 = 567 S_{4} +3S_{3} +27S_{2}=0 \implies S_{4}=567 S 5 + 3 S 4 + 27 S 3 = 0 ⟹ S 5 = − 7533 S_{5}+3S_{4} +27S_{3}=0 \implies S_{5}= -7533 S 6 + 3 S 5 + 27 S 4 = 0 ⟹ S 6 = 7290 S_{6}+3S_{5}+27S_{4}=0 \implies S_{6}=7290 S 7 + 3 S 6 + 27 S 5 = 0 ⟹ S 7 = 181521 S_{7}+3S_{6}+27S_{5}=0 \implies S_{7}=181521 which is the required value so n=7

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Jun Arro Estrella
Jan 20, 2016

Trial and error lol.

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