Find the greatest ter in the expansion of `(2+3x)^(9)` when `x=(3)/(2)`

Explanation

We have,
`(2+3x)^(9)=2^(9)(1+(3)/(2)x)^(9)`
In the expansion of `(1+(3)/(2)x)^(9)`, we have
`|(T_(r+1))/(T_(r))|=(9-r+1)/(r)|(3)/(2)x|=(10-r)/(r)|(3)/(2)xx(3)/(2)|`
`=(10-r)/(r)(9)/(4)=(90-9r)/(4r)`
`implies(T_(r+1))/(T_(r))ge1`
`implies(90-9r)/(4r)ge1`
`implies90ge13r`
`impliesr le(90)/(13)=6(12)/(13)=6+(12)/(13)`
`implies`Maximum value of r is 6.
Hence, the greatest term in the expansion of `(2+3x)^(9)`
`=2^(9)T_(6+1)` ltrgt `=2^(9).^(9)C_(6)((3)/(2)x)^(6)`
`=(9.8.7)/(1.2.3).(3^(12))/(2^(12))xx2^(9)`
`=(7)/(2)xx3^(13)`