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Question

Let [n2+1]=[n2+λ], where n,λN. Then λ can have y×n different values. Find y
(Note : [.] represents the greatest integer function.)

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Solution

We have,
n2+1=(n+1)22n<(n+1)2;nN
ie,
n2+1<n+1 or n<n2+1<n+1
[n2+1]=n
[n2+λ]=n
n<n2+λ<(n+1)
or n2<(n2+λ)<(n+1)2
0<λ<2n+1
Thus, λ can take 2n different values.
Hence y=2

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