To solve the problem of finding (a^7 + b^7) where (a) and (b) are roots of the quadratic equation (x^2 - 5x + 1 = 0), we use a recurrence relation for the sum of powers of roots.

Key Observations:

For roots (a) and (b) of (x^2 = 5x - 1):

• (a^n = 5a^{n-1} - a^{n-2}) and (b^n = 5b^{n-1} - b^{n-2}).
• Let (S_n = a^n + b^n), then the recurrence is:
  (Sn = 5S{n-1} - S_{n-2}).

Calculations:

• (S_0 = a^0 + b^0 = 2)
• (S_1 = a + b = 5) (sum of roots)
• (S_2 = 5S_1 - S_0 = 5×5 - 2 = 23)
• (S_3 = 5S_2 - S_1 = 5×23 - 5 = 110)
• (S_4 = 5S_3 - S_2 = 5×110 - 23 = 527)
• (S_5 = 5S_4 - S_3 = 5×527 - 110 = 2525)
• (S_6 = 5S_5 - S_4 = 5×2525 - 527 = 12098)
• (S_7 = 5S_6 - S_5 = 5×12098 - 2525 = 57965)

Answer: (\boxed{57965})

作者声明:本文包含人工智能生成内容。