Question

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.`

Answer 1

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`