Let be roots of x 2 + 2 x - 8 = 0 . If, then U 10 + 2 U 9 2 U 8 is equal to __________ . [2024]

STUDY MATERIALS
COURSES
MORE
SUCCESS STORY
ADMISSION
STORE

Back

JEE ADVANCE

CLASS XI - XII
CRASH COURSE

CLASS XI - XII

CRASH COURSE

JEE MAIN

CLASS XI - XII
CRASH COURSE

CLASS XI - XII

CRASH COURSE

NEET

CLASS XI - XII
CRASH COURSE

CLASS XI - XII

CRASH COURSE

BITSAT

CLASS XI - XII

CLASS XI - XII

CBSE BOARD

CLASS IX

CLASS X

CLASS XI

CLASS XII

CLASS VI

CLASS VII

CLASS VIII

UP BOARD

CLASS IX
CLASS X
CLASS XI
CLASS XII

CLASS IX

CLASS X

CLASS XI

CLASS XII

CUET

CRASH COURSE

CRASH COURSE

Courses

CLASSROOM COURSES
ONLINE COURSES

ONLINE COURSES

More

Solution

Q.
Let $α, β$ be roots of $x^{2} + \sqrt{2} x - 8 = 0 .$ If $U_{n} = α^{n} + β^{n},$ then $\frac{U_{10} + \sqrt{2} U_{9}}{2 U_{8}}$ is equal to __________ . [2024]

Ans.
(4)

Given that $α, β$ are roots of $x^{2} + \sqrt{2} x - 8 = 0$

$\Rightarrow α^{2} + \sqrt{2} α - 8 = 0$ and $β^{2} + \sqrt{2} β - 8 = 0$

$\Rightarrow α^{2} + \sqrt{2} α = 8$ and $β^{2} + \sqrt{2} β = 8$ ...(i)

Now, $\frac{U_{10} + \sqrt{2} U_{9}}{2 U_{8}} = \frac{α^{10} + β^{10} + \sqrt{2} α^{9} + \sqrt{2} β^{9}}{2 (α^{8} + β^{8})}$

$= \frac{α^{9} (α + \sqrt{2}) + β^{9} (β + \sqrt{2})}{2 (α^{8} + β^{8})}$

$= \frac{α^{9} (\frac{8}{α}) + β^{9} (\frac{8}{β})}{2 (α^{8} + β^{8})}$ [Using (i)]

$= \frac{8 (α^{8} + β^{8})}{2 (α^{8} + β^{8})} = 4$

Want Us to Email you About Special Offers & Updates?