1.

Ifthecoefficientsofx ^(9), x ^( 10) , x ^ ( 11 )intheexpansion of(1 +x ) ^nareinarithmeticprogressionthenn^ 2-41 n=

Answer»

`398`
`298`
`-398`
`198`

Solution :Theco- efficientsof` X ^9, x ^(10), x ^(11)`AREIN A.P
`therefore""^n c _9,""^n c _ (10),""^nc _(11)` arein `A.P`
`therefore2 (nc _ (10)) =nc _ 9 +nc _ (11)`
` rArr2= (nc _9 ) /(nc _ (10))+(nc _(11))/(nc _ (10)) `
` rArr 2= (10)/(n - 10+1 ) +(n - 11+ 1 )/(11)[ because(nc _r )/(nc _ (R - 1 ))= (n -r + 1 )/(r ) ] `
` rArr 2=(10)/(n - 9)+(n - 10)/(11)`
`rArr2-(10)/(n-9 )=(n -10)/(11)`
` rArr(2n -28)/(n -9 ) = (n -10)/(11)`
` rArr11 (2n- 28) =(n - 9 )(n - 10) `
`rArr22n- 308=n ^ 2-19n +90 `
`thereforen^ 2 - 41 n+398=0 `


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