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The coefficient of three consecutive terms in the expansion of (1+x)^n  are in the ratio 1 : 7 : 42, then 

Option: 1

n is divisible by 5


Option: 2

n is divisible by 7


Option: 3

 n is divisible by 12


Option: 4

n = 45


Answers (1)

Let the consecutive terms be rth, (r+1)th and (r+2)th  terms 

So, the condition leads to ^nC_{r-1}:^nC_r : ^n C_{r+1} = 1 : 7 : 42

\\\frac{^nC_{r}}{^nC_{r-1}}=7\\\frac{\frac{n!}{\left(n-r\right)!r!}}{\frac{n!}{\left(n-r+1\right)!\left(r-1\right)!}}=7\\\frac{n+1-r}{r}=7\\n+1=8r..................(1)

\\\frac{^nC_{r+1}}{^nC_{r}}=6\\\frac{\frac{n!}{\left(n-r-1\right)!\left(r+1\right)!}}{\frac{n!}{\left(n-r\right)!r!}}=6\\\frac{n-r}{r+1}=6\\n-6=7r..................(2)

  from (1)  and  (2)   

r = 7

and n = 55

Posted by

Kshitij

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