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The coefficients of three consecutive te...

The coefficients of three consecutive terms of `(1+x)^(n+5)` are in the ratio 5:10:14. Then `n=` ___________.

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Let the three consecutive terms in `(1+x)^(n+5)` be `t_(r), t_(r+1),t_(r+2)` having coefficient `""^(n+5)C_(r-1) , ""^(n+5)C_(r), ""^(n+5)C_(r+1)`
Given , `""^(n+5)C_(r-1) ,:""^(n+5)C_(r) : ""^(n+5)C_(r+1) =5:10:14`
`:. (""^(n+5)C_(r))/(""^(n+5)C_(r-1))=(10)/(5) and (""^(n+5)C_(r+1))/(""^(n+5)C_(r))=(14)/(10) `
`implies (n+5-(r-1))/(r)=2 and (n-r+5)/(r+1)=(7)/(5)`
`implies n-r+6 =2r`
` and 5n-5r+25=7r+7`
` implies n+6 =3r and 5n+18=12r `
`:. (n+6)/(3)=(5n+18)/(12)`
`implies 4n+24 =5n+ 18implies n=6 `
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