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If (1+x)^(n)=sum(r=0)^(n)a(r)x^(r) and b...

If `(1+x)^(n)=sum_(r=0)^(n)a_(r)x^(r) and b_(r)=1+(a_(r))/(a_(r-1)), and prod_(r=1)^(n)b_(r)=((101)^(100))/(100!)` then n equals:

Text Solution

Verified by Experts

Here, `a_(r) = ""^(n)C_(r)`
`therefore b_(r) = 1 +(a_(r))/(a_(1 -1)) = 1 + (""^(n)C_(r))/(""^(n)C_(r-1))`
`= 1 + (n-r+1)/r = (n+1)/r`
`rArr prod_(r=1)^(n) b_(r) = prod_(r=1)^(n) ((n+1))/r `
`= ((n+1))/1 cdot ((n+1))/2 cdot ((n+1))/3 ... ((n+1))/n= ((n+1)^(n))/(n!)`
`=((101)^(100))/(100!) ` [ giben]
`therefore n= 100 rArr n/20 = 5`
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