if-a-b-c-3-a-2-b-2-c-2-5-a-3-b-3-c-3-27-find-a-100-b-100-c-100-

Question Number 171749 by Mikenice last updated on 20/Jun/22

i f a + b + c = 3, a^{2} + b^{2} + c^{2} = 5, a^{3} + b^{3} + c^{3} = 27 . f i n d a^{100} + b^{100} + c^{100} .

Answered by floor(10²Eta[1]) last updated on 20/Jun/22

{(a + b + c)}^{2} = a^{2} + b^{2} + c^{2} + 2 (ab + ac + bc)

9 = 5 + 2 (ab + ac + bc) \Rightarrow ab + ac + bc = 2

(a^{2} + b^{2} + c^{2}) (a + b + c) = a^{3} + b^{3} + c^{3} + a^{2} b + a^{2} c + {ab}^{2} + b^{2} c + {ac}^{2} + {bc}^{2}

15 = 27 + ab (a + b) + ac (a + c) + bc (b + c)

- 12 = ab (3 - c) + ac (3 - b) + bc (3 - a)

- 12 = 3 (ab + ac + bc) - 3 abc

- 12 = 6 - 3 abc \Rightarrow abc = 6

let a, b, c be the roots of x^{3} + {Ax}^{2} + Bx + C

= (x - a) (x - b) (x - c)

= x^{3} - x^{2} (a + b + c) + x (ab + ac + bc) - abc

= x^{3} - {3 x}^{2} + 2 x - 6 = 0

x = 3 is a root

\Rightarrow x^{3} - {3 x}^{2} + 2 x - 6 = (x - 3) (x^{2} + mx + 2)

= x^{3} + (m - 3) x^{2} + (2 - 3 m) x - 6

\Rightarrow m - 3 = - 3 and 2 - 3 m = 2

m = 0

x^{3} - {3 x}^{2} + 2 x - 6 = (x - 3) (x^{2} + 2) = 0

x = 3, x^{2} = - 2 \Rightarrow x = \pm i \sqrt{2}

a = 3, b = i \sqrt{2}, c = - i \sqrt{2}

a^{100} + b^{100} + c^{100} = 3^{100} + 2^{50} + 2^{50} = 3^{100} + 2^{51}

Commented by Mikenice last updated on 20/Jun/22

t h a n k d s i r

Commented by Tawa11 last updated on 20/Jun/22

Great sir

Answered by mr W last updated on 20/Jun/22

l e t p_{n} = a^{n} + b^{n} + c^{n}

p_{1} = e_{1} = 3

p_{2} = e_{1} p_{1} - 2 e_{2} \Rightarrow 5 = 3^{2} - 2 e_{2} \Rightarrow e_{2} = 2

p_{3} = e_{1} p_{2} - e_{2} p_{1} + 3 e_{3} \Rightarrow 27 = 3 \times 5 - 2 \times 3 + 3 e_{3} \Rightarrow e_{3} = 6

p_{n} = e_{1} p_{n - 1} - e_{2} p_{n - 2} + e_{3} p_{n - 3}

p_{n} = 3 p_{n - 1} - 2 p_{n - 2} + 6 p_{n - 3}

r^{3} - 3 r^{2} + 2 r - 6 = 0

(r - 3) (r^{2} + 2) = 0

\Rightarrow r = 3, \pm i \sqrt{2} = \sqrt{2} e^{\pm \frac{π i}{2}}

p_{n} = A \times 3^{n} + 2^{\frac{n}{2}} (B \times e^{\frac{n π i}{2}} + C \times e^{- \frac{n π i}{2}})

w e g e t f r o m p_{1} = 3, p_{2} = 5, p_{3} = 27

A = B = C = 1

\Rightarrow p_{n} = 3^{n} + 2^{\frac{n}{2} + 1} \cos \frac{n π}{2}

\Rightarrow p_{100} = 3^{100} + 2^{50 + 1} \times \cos 50 π = 3^{100} + 2^{51}

\Rightarrow p_{101} = 3^{101} + 2^{\frac{103}{2}} \times \cos \frac{101 π}{2} = 3^{101}

\Rightarrow p_{102} = 3^{102} + 2^{51 + 1} \times \cos 51 π = 3^{102} - 2^{52}

Commented by Tawa11 last updated on 20/Jun/22

Great sir

Commented by floor(10²Eta[1]) last updated on 20/Jun/22

what is p_{1}, \dots and e_{1}, . .

Commented by mr W last updated on 20/Jun/22

s e e Q 74970

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