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The largest term in the expansion of (3+...

The largest term in the expansion of `(3+2x)^50`, where x=1/5, is

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We have,
`(T_(r+1))/(T_(r)) = (20-r+1)/(r). (5x)/(4) ge 1`
`rArr (20-r+1)/(r).(5)/(12) ge 1` " " (Putting `x=1//3`)
`rArr 105- 5r ge 12r`
`rArr 17r le 105`
`rArr r le 6'(3)/(17)` Therefore, for `r = 6` i.e., `7^(th)` term is the greatest term.
So, the second largest term may be `6^(th)` or `8^(th)`.
now, `(T_(8))/(T_(6)) = (.^(20)C_(7))/(.^(20)C_(5)).((5x)/(4))^(2)`
= `=((20!)/(13!7!))/((20!)/(15!15!))(5/12)^(2)`
`= (15 xx 14)/(7 xx 6) xx 5/12 xx 5/12 lt 1`
Thus, `T_(8) lt T_(6)`.
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