SATHEE: Principle Of Mathematical Induction Question 52

Principle Of Mathematical Induction Question 52

Question: If $ P=n(n^{2}-1^{2})(n^{2}-2^{2})(n^{2}-3^{2})…(n^{2}-r^{2}), $ $ n>r,n\in N $ then P is necessarily divisible by

Options:

A) $ (2r+2)! $

B) $ (2r+4)! $

C) $ (2r+1)! $

D) None of these

Show Answer

Answer:

Correct Answer: C

Solution:

[c] $ P=n(n+1)(n-1)(n+2)(n-2)…(n+r)(n-r). $ $ ={n(n+1)(n+2)…(n+r)}{(n-1)(n-2)…(n-r)} $ $ =(n+r)(n+r-1)…(n+1)(n)(n-1)…(n-r) $ Clearly P is product of $ (2r+1) $ consecutive integers, so divisible by $ (2r+1)! $

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Principle Of Mathematical Induction Question 52

Question: If $ P=n(n^{2}-1^{2})(n^{2}-2^{2})(n^{2}-3^{2})…(n^{2}-r^{2}), $ $ n>r,n\in N $ then P is necessarily divisible by

Options:

Answer:

Solution:

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