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Question

Statement I For every natural number n2
11+12++1n>n
Statement II  For every natural number n2
                   n(n+1)<n+1


A
Statement I is true, Statement II is true; and Statement II is correct explanation for Statement I.
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B
Statement I is true, Statement II is true; and Statement II is not correct explanation for Statement I.
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C
Statement I is true, Statement II is false.
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D
Statement I is false, Statement II is true.
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Solution

The correct option is C Statement I is true, Statement II is true; and Statement II is correct explanation for Statement I.
For every natural number n, we have 
n(n+1)=n2+n<n2+n+n+1
n(n+1)<(n+1)2
n(n+1)<(n+1)  n2
Statement II is true.
Also, from above, we have
n<n+1
1n>1n+1   n2
11>12>13>>1n1>1n   n2
11>1n
12>1n
13>1n
    
    
1n=1n,n2
Adding all, we get
11+12+13++1n>nn=n
Statement I is true.

Mathematics

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