# Let c and b be the roots of x2-6x-2=0 (cb). Ifan=cn-bn where (n=1)then a10-2a8/2a9 = 1 2 3 4

c and b are the roots of the equation : ............(1)
if
c is the root of eq(1): therefore
${c}^{2}-6c-2=0\phantom{\rule{0ex}{0ex}}⇒{c}^{2}=6c+2.........\left(2\right)$
multiplying through out by  ${c}^{n-2}$ on the both sides:
${c}^{n}=6{c}^{n-1}+2{c}^{n-2}.........\left(3\right)$
similarly ${b}^{n}=6{b}^{n-1}+2{b}^{n-2}..........\left(4\right)$
subtracting eq(4) by eq(3):
${c}^{n}-{b}^{n}=6\left({c}^{n-1}-{b}^{n-1}\right)+2\left({c}^{n-2}-{b}^{n-2}\right)\phantom{\rule{0ex}{0ex}}{a}_{n}=6{a}_{n-1}+2{a}_{n-2}........\left(5\right)$
now put n = 10
${a}_{10}=6{a}_{9}+2{a}_{8}$
therefore ${a}_{10}-2{a}_{8}=6{a}_{9}$
hence
$\frac{{a}_{10}-2{a}_{8}}{2{a}_{9}}=\frac{6{a}_{9}}{2{a}_{9}}=3$
hope this helps you

• 124
What are you looking for?