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If `a_1,a_2, a_3, a_4` be the coefficient of four consecutive terms in the expansion of `(1+x)^n ,` then prove that: `(a_1)/(a_1+a_2)+(a_3)/(a_3+a_4)=(2a_2)/(a_2+a_3)dot`

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Let the coefficients of `T_(r),T_(r+1),T_(r+2),T_(r+3)` be `a_(1),a_(2),a_(3),a_(4)`, respectively in the expansion of `(1+x)^(n)`. Then,
`(a_(2))/(a_(1)) = (.^(n)C_(r))/(.^(n)C_(r+1)) = (n-r+1)/(r )`
or `1+(a_(2))/(a_(1)) = (n+1)/(r)`
Similarly
`1+(a_(3))/(a_(2)) = (n+1)/(r+1)` and `1+(a_(4))/(a_(3)) = (n+1)/(r+2)`
Now,
`L.H.S.=(a_(1))/(a_(1)+a_(2))+(a_(3))/(a_(3)+a_(4))=(1)/(1+(a_(2))/(a_(1)) )+(1)/(1+(a_(4))/(a_(3)))`
`=(r)/(n+1)+(r+2)/(n+1)=(2(r+1))/(n+1)=2(1)/(1+(a_(3))/(a_(2)))= (2a_(2))/(a_(2)+a_(3))`
`=R.H.S.`
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