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

In the coefficients of rth, `(r+1)t h ,a n d(r+2)t h` terms in the binomial expansion of `(1+y)^m` are in A.P., then prove that `m^2-m(4r+1)+4r^2-2=0.`

Text Solution

Verified by Experts

Here coefficient of `T_(r ), T_(r+1)`, and `T_(r+2)` in `(1+y)^(m)` are in
A.P.
`rArr .^(m)C_(r-1), .^(m)C_(r )` and `.^(m)C_(r+1)` are in A.P.
`rArr 2.^(m)C_(r ) = .^(m)C_(r-1)+.^(m)C_(r+1)`.
or `2 = (.^(m)C_(r+1))/(.^(m)C_(r )) + (.^(m)C_(r-1))/(.^(m)C_(r )) = (r)/(m-r+1)+(m-r)/(r+1)`
or `2-(r)/(m-r+1)=(m-r)/(r+1)`
or `(2m-3r+2)/(m-r+1) = (m-r)/(r+1)`
or `2mr -3r^(2) +2r+2m-3r+2 = m^(2) - mr + m - mr + r^(2) - r`
or `m^(2)(4r+1)m + 4r^(2) - 2 = 0`
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 the coefficients of r^(th),(r + 1)^(th) and (r + 2)^(th) terms in the binomial expansion of (1 + y)^(m) are in A.P., then m and r satisfy the equation

If the coefficients of the rth, (r+1)t h ,(r+2)t h terms is the expansion of (1+x)^(14) are in A.P, then the largest value of r is.

If the coefficients of (r-5)^th and (2r - 1)^th terms in the expansion of (1 + x)^34 are equal, find r.

If the coefficient of the 2^(nd) , 3^(rd) and 4^(th) terms in the expansion of (1 + x)^(n) are in A.P, then n=

If the coefficients of the (2r+4)t h ,(r-2)t h term in the expansion of (1+x)^(18) are equal, then the value of r is.

If the coefficients of a^r-1 , a^r and a^r+1 in the expansion of (1+a)^n are in arithmetic progression, prove that n^2 - n(4r+1)+ 4r^2 - 2 =0.

If the coefficients of the r^(th) term and the (r + 1)^(th) term in the expansion of (1 + x)^(20) are in the ratio 1:2 , then r is equal to

If the coefficients of rth and (r+1)t h terms in the expansion of (3+7x)^(29) are equal, then r equals a. 15 b. 21 c. 14 d. none of these

If r^[th] and (r+1)^[th] term in the expansion of (p+q)^n are equal, then [(n+1)q]/[r(p+q)] is