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.

Answer

4

Explanation

Coefficient of `r^(th)` and `(r+2)^(th)` terms are .^(14)C_(t-1),^(14)C_(r)` and .^(14)C_(r+1)`
If these coefficients are in A.P. then `2(` .^(14)C_(r))=^(14)C_(r-1)+^(14)C_(r+1)`
`implies(2(14)!)/(r!(14-r)!)=((14)!)/((r-1)!(15-r)!)+((14)!)/((r+1)!(13-r)!)`
`implie3s(2(14)!)/(r!(14-r)!)+((14)![(r+1)r+(15-r)(14-r)])/((r+1)!(15-r)!)`
`implies2(15-r)(r+1)=2r^(2)-28r+210`
`impliesr^(2)-14r+45=0`
`implies(r-5)(r-9)=0`
`impliesr=5` or `9`