Home
Class 12
MATHS
If the coefficients of rth, (r+1)t h ,a ...

If the coefficients of rth, `(r+1)t h ,a n d(r+2)t h` terms in the expansion of `(1+x)^(14)` are in A.P., then `r` is/are a. 5 b. 11 c. `10` d. `9`

A

5

B

9

C

10

D

12

Text Solution

Verified by Experts

The correct Answer is:
a,b
Coefficient of rth (r+1) th and (r +2) th terms in ` ( 1 + x)^(14)` are
` ""^(14)C_(r-1), ""^(14)C_(r)`
and ` ""^(14)C_(r-1)`,respectively
Now , according to the question , ` 2 (""^(14)C_(r)) = ""^(14)C_(r-1) + ""^(14)C_(r+1)`
On dividing both sides by ` ""^(14)C_(r-1) ` , we get
`2= (""^(14)C_(r-1))/(""^(14)C_(r)) = (""^(14)C_(r+1))/(""^(14)C_(r))`
` rArr 2= (r)/(14 - r + 1) + (14 - (r +1)+1)/(r +1)`
` rArr 2 = (r)/(15-r) + (14-r)/(r+1)`
` rArr 2(15-r) (r+1)= r(r+1) + (15 - r) (14 - r)`
` rArr - 2r^(2) + 28r + 30 + 2r^(2) - 28 r + 210`
` rArr 4t^(2) - 56r + 180 = 0 rArr r^(2) - 14r + 45 = 0 `
` rArr (r-9)(r-5) = 0 `
` rArr r = 5,9`
Promotional Banner

Topper's Solved these Questions

  • BIONOMIAL THEOREM

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

  • BIONOMIAL THEOREM

    ARIHANT MATHS|Exercise Binomial Theorem Exerciese 4 : Single Integer Answer Type Questions|1 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

If the coefficients of 5th, 6th , and 7th terms in the expansion of (1+x)^n are in A.P., then n=

If the coefficient of the rth, (r+1)th and (r+2)th terms in the expansion of (1+x)^(n) are in A.P., prove that n^(2) - n(4r +1) + 4r^(2) - 2=0 .

Find the value of r if the coefficients of (2r+4)th and (r - 2) th term in the expansion of (1+ x)^18 are equal.

The coefficients of the (r-1)^(th) , r^(th) and (r+1)^(th) terms in the expansion of (x + 1)^n are in the ratio 1:3:5. Find n and r.

If the rth term in the expansion of (x/3-2/x^2)^10 contains x^4 , then r is equal to ,

If the coefficients of a^(r-1), a^(r ), a^(r+1) in the binomial expansion of (1 + a)^(n) are in Arithmetic Progression, prove that: n^(2) -n(4r+1) + 4r^(2) -2=0

If the greatest term in the expansion of (1+x)^(2n) has the greatest coefficient if and only if xepsilon(10/11, 11/10) and the fourth term in the expansion of (kx+ 1/x)^m is n/4 then find the value of mk.

In the binomial expansion of (1+x)^34 , the coefficients of the (2r-1)th and the (r-5)th terms are equal. Find r.