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Question

The coefficient of (r1)th, (r)th, and (r+1)th terms in the expansion of (r1)th are in the ratio 1:3:5. Find n and r.

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Solution

We have,

coefficientof(r1)thtermscoefficientofrthterms=13

nCr2nCr1=13


[n!(r2)!(nr+2)!][n!(r1)!(nr+1)!]=13


(r1)!(nr+1)!(r2)!(nr+2)!=13


(r1)(r2)!(nr+1)!(r2)!(nr+2)(nr+1)!=13


(r1)(nr+2)=13

3r3=nr+2

4r=n+5 ……… (1)

Similarly,

coefficientofrthtermscoefficientof(r+1)thterms=35

nCr1nCr=35


n!(r1)!(nr+1)!n!r!(nr)!=35


r!(nr)!(r1)!(nr+1)!=35


r(r1)!(nr)!(r1)!(nr+1)(nr)!=35


r(nr+1)=35

5r=3n3r+3

8r=3n+3 ………. (2)

From equation (1) and (2), we get

n=7,r=3


Hence, the value of n,r=7,3, respectively.


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