Official solution
\[\alpha \] and
\[\beta \] be the roots of the equation
5x2 + 6x -- 2 = 0.
\[ \Rightarrow \] 5\[\alpha \]2 + 6\[\alpha \] - 2 = 0
\[ \Rightarrow \] 5\[\alpha \]n + 2 + 6\[\alpha \]n + 2 - 2\[\alpha \]n = 0 ......(1)
(By multiplying \[\alpha \]n)
Similarly 5\[\beta \]n + 2 + 6\[\beta \]n + 2 - 2\[\beta \]n = 0 ......(2)
By adding (1) \& (2)
5Sn+2 + 6Sn+1 -- 2Sn = 0
For n = 4
5S6
+ 6S5
= 2S4