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Square Sum Settlement (Posted on 2015-01-14) Difficulty: 2 of 5
If x2 + x +1 =0, then find the value of:
(x+1/x)2 + (x2 + x-2)2 + ...+ (x9 + x-9)2

No Solution Yet Submitted by K Sengupta    
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Some Thoughts Avoiding complexities.. (spoiler) Comment 3 of 3 |

(xn+1 + x-(n+1)) = (xn + x-n)(x + x-1) – (xn-1 + x-(n-1)),

so writing fn = xn + x-n  gives:      fn+1 = fn*f1 - fn-1     (1)

x2 + x + 1 = 0 =>  x(x + x-1) = -x and, since x = 0 is

not a solution, we can divide by x to obtain  f1 = -1,

then (1) becomes:   fn+1 = -(fn + fn-1).

From the definition, f0 = 2, so this Fibonacci-like

recurrence equation can be used to give

f0, f1, …….. = 2, -1, -1, 2, -1, -1, 2, -1, -1, 2……

and the required Sum1 to 9(fn2) is:

1 + 1 + 4 + 1 + 1 + 4 + 1 + 1 + 4 = 18



  Posted by Harry on 2015-01-15 08:20:20
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