The expression \frac{2^2+1}{2^2-1}+\frac{3^2+1}{3^2-1} +\frac{4^2+1}{4^2-1}+.... +\frac{2011^2+1}{2011^2-1} lies in the interval

Correct Answer is : (2011,201112)

KVPY SA Exam 2011 Question Paper

Question : 62 of 80

Marks: +1, -0

The expression

{2^2+1}/{2^2-1}+{3^2+1}/{3^2-1}

+{4^2+1}/{4^2-1}+....

+{2011^2+1}/{2011^2-1}

lies in the interval

Solution:

{2^2+1}/{2^2-1}+{3^2+1}/{3^2-1}

+{4^2+1}/{4^2-1}+....

+{2011^2+1}/{2011^2-1}

∑↙{r=2}↖{2011}{r^2+1}/{r^2-1}

=∑↙{r=2}↖{2011}[1+2/{(r+1)(r-1)}]

=∑↙{r=2}↖{2011}[1+1/{r-1}-1/{r+1}]

=2010+[1-1/3+1/2

-1/4+1/3-1/5

+...+1/2010-1/2012]

=2010+1+1/2-1/2012-1/2011

=2011+1/2-[1/2011+1/2012]

lies between

(2011,2011 1/2)

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