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|Exercise Exercise 8.1|17 Videos

  • BINOMIAL THEOREM

    CENGAGE|Exercise Exercise 8.2|10 Videos

  • AREA UNDER CURVES

    CENGAGE|Exercise Question Bank|10 Videos

  • CIRCLE

    CENGAGE|Exercise MATRIX MATCH TYPE|6 Videos

Similar Questions

Explore conceptually related problems

If I_(n)=int_(0)^(pi/2) sin^(x)x dx , then show that I_(n)=((n-1)n)I_(n-2) . Hence prove that I_(n)={(((n-1)/n)((n-3)/(n-2))((n-5)/(n-4))………(1/2)(pi)/2,"if",n"is even"),(((n-1)/n)((n-3)/(n-2))((n-5)/(n-4))………(2/3)1,"if",n"is odd"):}

Prove that n! ( n +2) = n ! + ( n + 1) !

If C_(r ) = (n!)/(r!(n - r)!) , then prove that sqrt(C_(1)) + sqrt(C_(2)) + …. + sqrt(C_(n)) sqrt(n(2^(n) - 1))

If u_n=sin^("n")theta+cos^ntheta, then prove that (u_5-u_7)/(u_3-u_5)=(u_3)/(u_1) .

Prove that: ((2n)!)/(n !)={1. 3. 5 (2n-1)}2^ndot

If A=[(3,-4),(1,-1)] , then prove that A^(n)=[(1+2n,-4n),(n,1-2n)] , where n is any positive integer.

If z_1a n dz_2 are complex numbers and u=sqrt(z_1z_2) , then prove that |z_1|+|z_2|=|(z_1+z_2)/2+u|+|(z_1+z_2)/2-u|

If S_n=[1/(1+sqrt(n))+1/(2+sqrt(2n))+....+1/(n+sqrt(n^2))] then (lim)_(n ->oo)S_n is equal to (A) log 2 (B) log4 (C) log8 (D) none of these

IfI_(m , n)=int_0^(pi/2)sin^m xcos^n xdx , Then show that I_(m , n)=(m-1)/(m+n)I_m-2n(m ,n in N) Hence, prove that I_(m , n)=f(x)={((n-1)(n-3)(m-5)(n-1)(n-3)(n-5))/((m+n)(m+n-2)(m+n-4))pi/4 when both m and n are even ((m-1)(m-3)(m-5)(n-1)(n-3)(n-5))/((m+n)(m+n-2)(m+n-4))

If U_n=int_0^pi(1-cosnx)/(1-cosx)dx , where n is positive integer or zero, then show that U_(n+2)+U_n=2U_(n+1)dot Hence, deduce that int_0^(pi/2)(sin^2ntheta)/(sin^2theta)=1/2npidot