Question

急求,m平方=m+1,n平方=n+1.求m五次方+n五次方=

Answer 1

m^5 + n^5 = m^3 * m^2 + n^3 * n^2 = m^3 * (m+1) + n^3 * (n+1) = m^4 + m^3 + n^4 + n^3 = (m+1)^2 + m(m+1) + (n+1)^2 + n(n+1) = m^2 + 2m + 1 + m^2 + m + n^2 + 2n + 1 + n^2 + n = 2m^2 + 3m + 1 + 2n^2 + 3n + 1 = 2(m+1) + 3m + 1 + 2(n+1) + 3n + 1 = 2m + 2 + 3m + 1 + 2n + 2 + 3n + 1 = 5m + 3 + 5n + 3 = 5m + 5n + 6 = 5(m+n) + 6 因为m^2 = m + 1,n^2 = n + 1,则m,n为方程 x^2 = x + 1的两个根,则有韦达定理,两根之和 m + n = - b/a = 1,所以5(m+n) + 6 = 5 * 1 + 6 = 11.所以 m^5 + n^5 = 5(m+n) + 6 = 5 * 1 + 6 = 11