Are you smarter than me?

$\frac { 1 }{ 1+{ 2 }^{ \sqrt { 1947 } } } +\frac { 1 }{ { 2 }^{ 1947 }+{ 2 }^{ \sqrt { 1947 } } }$

$=\frac { 1 }{ 1+{ 2 }^{ \sqrt { 1947 } } } +\frac { 1 }{ 1+{ 2 }^{ \sqrt { 1947 } } } \left( \frac { 1 }{ { 2 }^{ \sqrt { 1947 } } } \right)$

$=\left( 1+\frac { 1 }{ { 2 }^{ \sqrt { 1947 } } } \right) \left( \frac { 1 }{ 1+{ 2 }^{ \sqrt { 1947 } } } \right)$

$=\left( \frac { { 2 }^{ \sqrt { 1947 } }+1 }{ { 2 }^{ \sqrt { 1947 } } } \right) \left( \frac { 1 }{ 1+{ 2 }^{ \sqrt { 1947 } } } \right)$

$=\frac { 1 }{ { 2 }^{ \sqrt { 1947 } } }$

Similarly pairing remaining terms (Note that there is no term without a pair), we get,

$\sum _{ n=0 }^{ 1947 }{ \frac { 1 }{ { 2 }^{ n }+\sqrt { { 2 }^{ 1947 } } } }$

= $\frac { 974 }{ { 2 }^{ \sqrt { 1947 } } }$

This is what I did in the exam but didn't mark the option as there was no option matching to the above answer. And I got home and realised what a fool I was, FORGOT to cancel common factor 2. AGHHH!!!!!!!!

So, the answer is

= $\boxed{\frac {487 }{ { 2 }^{ \sqrt { 1945 } } }}$

Need help in LaTeX to make text bigger.

this is the second question of part 2 of SA stream KVPY exam 2014 and i solved it correctly.

Just add the first-last term,second-second last and so on U WOULD OBSERVE A MAGIC Can anyone guess