Home
Class 12
MATHS
If Un=(sqrt(3)+1)^(2n)+(sqrt(3)-1)^(2n) ...

If `U_n=(sqrt(3)+1)^(2n)+(sqrt(3)-1)^(2n)` , then prove that `U_(n+1)=8U_n-4U_(n-1)dot`

Text Solution

Verified by Experts

`U_(n) = [(sqrt(3) + 1)^(2)]^(n) + [(sqrt(3) - 1)^(2)]^(n)`
`= (4+2sqrt(3))^(n) + (4-2sqrt(3))^(n)`
` = alpha^(n) + beta^(n)` where `alpha + beta = 8, alphabeta = 4`
Now, `8Y_(n) = (alpha+beta)(alpha^(n)+beta^(n))`
`= alpha^(n+1)+beta^(n+1)+betaalpha^(n)`
`= U_(n+1)+alphabeta(alpha^(n-1)+beta^(n-1))`
`= U_(n+1)+4U_(n-1)`
`rArr U_(n+1) = 8U_(n) - 4U_(n-1)`.
Promotional Banner

Topper's Solved these Questions

  • BINOMIAL THEOREM

    CENGAGE PUBLICATION|Exercise Concept Application Exercise 8.1|17 Videos

  • BINOMIAL THEOREM

    CENGAGE PUBLICATION|Exercise Concept Application Exercise 8.2|10 Videos

  • BINOMIAL THEOREM

    CENGAGE PUBLICATION|Exercise Comprehension|11 Videos

  • AREA

    CENGAGE PUBLICATION|Exercise Comprehension Type|2 Videos

  • CIRCLE

    CENGAGE PUBLICATION|Exercise For problems 3 and 4|2 Videos

Similar Questions

Explore conceptually related problems

If n is a positive integer and U_(n) = (3 + sqrt5)^(n) + (3 - sqrt5)^(n) , then prove that U_(n + 1) = 6U_(n) - 4U_(n -1), n ge 2

Prove that [(n+1)//2]^n >(n !)dot

If u_(n)=sin^(n)alpha+cos^(n)alpha , then prove that 2u_(6)-3u_(4)+1=0 .

If U_n=2 cosntheta then U_1U_n-U_(n-1) is equal to

If n is a positive integer , then (sqrt(3)+1)^(2n)-(sqrt(3)-1)^(2n) is -

If (1+x)^n=sum_(r=0)^n C_r x^r , then prove that C_1+2C_2+3C_3+....+n C_n=n2^(n-1)dot .

If n is a possible integer, then (sqrt3+1)^(2n)-(sqrt3-1)^(2n) is

If sqrt(1-x^(2n))+sqrt(1-y^(2n))=a^(n)(x^(n)-y^(n)) , prove that, (dy)/(dx)=((x)/(y))^(n-1).sqrt((1-y^(2n))/(1-x^(2n))

If u_(n)=2 cos n theta , then show that u_(n+1)= u_(1)u_(n)-u_(n-1) , hence, show that 2 cos 5 theta= u_(1)^(5)- 5u_(1)^(3)+ 5u_(1)