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Grade 12Algebra

The greatest positive integer which divides (n+1)(n+2)(n+3) upto (n+r) forall n belongs to N is
a)r
b)r!
c)(r+1)!
d)n+r
I think option c) is correct option. This from the chapter mathematical induction.But Ravisudhakar sir said option b) is correct.Please explain in detail.

Profile image of Raja Sekhar Reddy Eragamreddy
8 Years agoGrade 12
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1 Answer

Profile image of Arun
8 Years ago
 
Dear student
 
P(n) = (n+1)(n+2).........(n+r)
 
Now find p(1)
P(1) = r! (r+1)
 
Similarly
P(k) will be divisible by r!
P(k+1) is also divisible by r!
 
Hence p(n) qill also be divisible by r!.