Home
Class 12
MATHS
Let (1 + x^(2))^(2) (1 + x)^(n) = a(0) ...

Let ` (1 + x^(2))^(2) (1 + x)^(n) = a_(0) + a_(1) x + a_(2) x^(2) + …` if
` a_(1),a_(2) " and " a_(3)` are in A.P , the value of n is

A

2

B

3

C

4

D

7

Text Solution

Verified by Experts

The correct Answer is:
b,c
We have ,
Coefficient of ` x , x^(2) " and " x^(3) " in " (1 + x^(2))^(2) (1 + x)^(n)`
i.e. , values of ` a_(1) , a_(2) " and " a_(3) " in " (1 + 2x^(2) + x^(4) ) (1 + x)^(n)`
` rArr a_(1) = ""^(n)C_(1) , a_(2) = ""^(n)C_(2) + 2 " and " a_(3) = ""^(n)C_(3) + 2 ""^(n)C_(1)`
According to the question ,
` 2a_(2) = a_(1) + a_(3)`
`rArr 2(""^(n)C_(2)) = ""^(n)C_(1) + (""^(n)C_(3) + 2^(n)C_(1))`
` rArr [2(n(n-1))/(2) ] + 4 = 3n + (n(n-1)(n-2))/(6)`
` rArr n^(3) - 9n^(2) = 26n = 24 = 0 `
` rArr (n - 2) (n^(2) - 7n + 12) = 0 `
` rArr (n-2)(n-3) (m-4)= 0 `
` rArr n = 2,3,4`
Hence , n == 3,4,(` n ne 2 because ""^(n)C_(3)` is not defined )
Promotional Banner

Topper's Solved these Questions

  • BIONOMIAL THEOREM

    ARIHANT MATHS|Exercise Exercise (Passage Based Questions)|21 Videos

  • BIONOMIAL THEOREM

    ARIHANT MATHS|Exercise Exercise (Single Integer Answer Type Questions)|9 Videos

  • BIONOMIAL THEOREM

    ARIHANT MATHS|Exercise Exercise (Single Option Correct Type Questions)|30 Videos

  • AREA OF BOUNDED REGIONS

    ARIHANT MATHS|Exercise Exercise (Questions Asked In Previous 13 Years Exam)|23 Videos

  • CIRCLE

    ARIHANT MATHS|Exercise Exercise (Questions Asked In Previous 13 Years Exam)|16 Videos

Similar Questions

Explore conceptually related problems

Let (1+x^(2))^(2)*(1+x)^(n)=sum_(k=0)^(n+4)a_(k)*x^(k) If a_(1),a_(2) and a_(3) are iun AP, find n.

Given (1+x^(2))^(2)(1+x)^(n)=sum a_(r)x^(r). If a_(0),a_(1) and a_(2) are in A.P.then value of n can be:

(1+x^(2))^(2)(1+x)^(n)=A_(0)+A_(1)x+A_(2)x^(2)+... If A_(0),A_(1),A_(2) are in A.P.then the value of n is-

If (1 + x + x^(2) + x^(3))^(n)= a_(0) + a_(1)x + a_(2)x^(2) + a_(3) x^(3) +...+ a_(3n) x^(3n) , then the value of a_(0) + a_(4) +a_(8) + a_(12)+….. is

If (1 + x+ 2x^(2))^(20) = a_(0) + a_(1) x + a_(2) x^(2) + …+ a_(40) x^(40) . The value of a_(0) + a_(2) + a_(4) + …+ a_(38) is

Let n in N . If (1+x)^(n)=a_(0)+a_(1)x+a_(2)x^(2)+…….+a_(n)x^(n) and a_(n)-3,a_(n-2), a_(n-1) are in AP, then :

If n ge 2 , and (1 + x + x^(2))^(n) = a_(0) + a_(1)x + a_(2)x^(2) + .. . +a_(2n) x^(2n) , then a_(0) - 2a_(1)+ 3a_(2) + ... + (2n + 1)a_(n) = ______