Question

Find the value of ${(a+\sqrt{a^2-1})}^7+{(a-\sqrt{a^2-1})}^7$

Answer 1

We have the sum of two expansions whose terms are numerically the same; but in the second expansion the second, fourth, sixth and eighth terms are negative, and therefore destroy corresponding terms of the first expansion.

$\therefore$ we get it equal to

$=2[a^7+21a^5(a^2-1)+35a^3(a^2-1{)}^2+7a(a^2-1)3]$

$=2a(64a^6-112a^4+56a^2-7)$