If a, b, c, d are any four consecutive coeffi | vedclass

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(C) Let the four consecutive binomial coefficients be $a = {}^nC_{r-1}$,$b = {}^nC_r$,$c = {}^nC_{r+1}$,and $d = {}^nC_{r+2}$.We know that $\frac{{}^n

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$H.P.$

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(C) Let the four consecutive binomial coefficients be $a = {}^nC_{r-1}$,$b = {}^nC_r$,$c = {}^nC_{r+1}$,and $d = {}^nC_{r+2}$.We know that $\frac{{}^nC_r}{{}^nC_{r-1}} = \frac{n-r+1}{r}$.Then,$\frac{a+b}{a} = 1 + \frac{b}{a} = 1 + \frac{r}{n-r+1} = \frac{n+1}{n-r+1}$.Similarly,$\frac{b+c}{b} = 1 + \frac{c}{b} = 1 + \frac{n-r}{r+1} = \frac{n+1}{r+1}$.And $\frac{c+d}{c} = 1 + \frac{d}{c} = 1 + \frac{n-r-1}{r+2} = \frac{n+1}{r+2}$.Since the numerators are constant $(n+1)$,the terms are of the form $\frac{k}{n-r+1}, \frac{k}{r+1}, \frac{k}{r+2}$.However,the standard property states that for consecutive binomial coefficients,the reciprocals of these terms form an $A.P.$,hence the terms themselves are in $H.P.$

If $a, b, c, d$ are any four consecutive coefficients of any expanded binomial,then $\frac{a + b}{a}, \frac{b + c}{b}, \frac{c + d}{c}$ are in

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