An algebra problem by swastik p (3)
If 2 0 1 1 + 2 0 1 1 2 − 1 1 = m − n , where m , n ∈ Z + , what is the value of m + n ?
This problem is a part of Problems for everyone!!!
This section requires Javascript.
You are seeing this because something didn't load right. We suggest you, (a) try
refreshing the page, (b) enabling javascript if it is disabled on your browser and,
finally, (c)
loading the
non-javascript version of this page
. We're sorry about the hassle.
2 solutions
2 0 1 1 + 2 0 1 1 2 − 1 = 2 0 1 1 + 2 0 1 0 × 2 0 1 2 = ( 1 0 0 5 ) 2 + ( 1 0 0 6 ) 2 + 2 1 0 0 5 1 0 0 6 = ( 1 0 0 5 + 1 0 0 6 ) 2 = 1 0 0 5 + 1 0 0 6 2 0 1 1 + 2 0 1 1 2 − 1 1 = 1 0 0 5 + 1 0 0 6 1 = ( 1 0 0 6 + 1 0 0 5 ) ( 1 0 0 6 − 1 0 0 5 ) 1 0 0 6 − 1 0 0 5 = 1 0 0 6 − 1 0 0 5 = m − n m + n = 1 0 0 6 + 1 0 0 5 = 2 0 1 1
square both sides, 1/(2011+√(〖2011〗^2-1)) = m+n - 2√mn 1/(2011+√(〖2011〗^2-1)) . (2011-√(〖2011〗^2-1))/(2011-√(〖2011〗^2-1)) = 2011-√((2012)(2010)), m+n = 2011, and 4mn=(2012)(2010), or mn = 503 . 2010 = 1006 . 1005