I get 989/32
2ab = -1
ab = - (1/2)
(ab)^2 = 1/4
(a^2 + b^2)^2 = a^4 + b^4 + 2a^2b^2 = 4
a^4 + b^4 + (1/2) = 4
a^4 + b^4 = 3.5
(a+b)(a^2 + b^2) = a^3 + b^3 + ab(a+b) = 2
a^3 + b^3 + (-1/2)(1) = 2
a^3 + b^3 = 5/2
(a^3 + b^3)^2 = 25/4
a^6 + b^6 + 2(ab)^3 = 25/4
a^6 + b^6 + 2(-1/8) = 25/4
a^6 + b^6 = 26/4 = 13/2
(a^2 + b^2)(a^3 + b^3) = 2*(5/2) = 5
(a^5 + b^5) + a^2b^2(a+b) = 5
(a^5 + b^5) + (1/4)(1) = 5
(a^5 + b^5) = 19/4
(a^5 + b^5)(a^6 + b^6) = 19/4 * 13/2 = 247/8
(a^11 + b^11) + (a^5*b^5)(a+b) = 247/8
(a^11 + b^11) + (- 1/32)(1) = 247/8
(a^11 + b^11) = 247/8 + 1/32 = 989/32
(a^11 + b^11) = 989/32
double check:
a+b=1
2ab = -1
b = -1/(2a)
a -1/(2a) =1
2a^2 - 1 = 2a
2a^2 - 2a - 1 = 0
a = (2 ± sqrt(12))/4 = (1 ± sqrt(3))/2
a is one, b is the other
a = (1 + sqrt(3))/2
b = (1 - sqrt(3))/2
and calculating 32*(a^11 + b^11) I get 988.9999999999997, close enough
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Posted by Larry
on 2023-03-22 11:01:38 |
Solution
