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Find ax5+by5 given that:

ax +by = 3

ax2+by2= 7

ax3+by3=16

ax4+by4=42

(axn+byn)(x+y)=(axn+1+byn+1)+(xy)(axn-1+byn-1)

*this is neither a problem i created nor my homework

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Find ax5+by5 given that:

ax +by = 3

ax2+by2= 7

ax3+by3=16

ax4+by4=42

(axn+byn)(x+y)=(axn+1+byn+1)+(xy)(axn-1+byn-1)

*this is neither a problem i created nor my homework

from the hint, let n = 2

7*(x+y) = 16+3*x*y

and n = 3

16*(x+y) = 42 + 7*x*y

These two equations have solutions

-7-sqrt(87) and sqrt(87) - 7

where x is one of these and y is the other.

z = ax^5 + bx^5

42*(x+y) = z + x*y*16

z = (x+y)*42 - 16*x*y = (-14)*42 -16*(-7-sqrt(87))*(-7+sqrt(87))

= -14*42 - 16*(49-87) = 20

btw:

a = 5.864681464178368e-05

b = 1.289415037395884

Edited by mmiguel1
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from the hint, let n = 2

7*(x+y) = 16+3*x*y

and n = 3

16*(x+y) = 42 + 7*x*y

These two equations have solutions

-7-sqrt(87) and sqrt(87) - 7

where x is one of these and y is the other.

z = ax^5 + bx^5

42*(x+y) = z + x*y*16

z = (x+y)*42 - 16*x*y = (-14)*42 -16*(-7-sqrt(87))*(-7+sqrt(87))

= -14*42 - 16*(49-87) = 20

absolutely!

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