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Find the greatest coefficient in the expansion of `(1+2x//3)^(15)dot` .

Text Solution

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The greatest coefficient is equal to the greatest term when `x = 1`.
For `x = 1, (T_(r+1))/(T_(r)) (15-r+1)/(r ) xx 2/3`
Let `(T_(r+1))/(T_(r)) ge 1`
or `(15-r+1)/(r ) 2/3 ge 1`
or `32-2r ge 3r`
or `r le 32//5`
So, for greatest coefficient, `r =6`.
Hence, `7^(th)` term has the greatest coefficient and its value is `.^(15)C_(6)(2//3)^(6)`.
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