Rút gọn biểu thức A

To simplify the expression, we can first calculate each term individually and then add them all together.

a = (2^2)/(1*3) + (3^2)/(2*4) + (4^2)/(3*5) + ... + (2011^2)/(2010*2012)

a = 4/3 + 9/8 + 16/15 + ... + 4044121/4044120

To find a common denominator for all the fractions, we can multiply each fraction by the product of all the denominators:

a = (4*8*15*...*2010*2012 + 9*3*15*...*2010*2012 + 16*3*8*...*2010*2012 + ... + 4044121*3*8*...*2010)/(3*8*15*...*2010*2012)

Now we can simplify the numerator by expanding each term:

a = (4*8*15*...*2010*2012 + 9*3*15*...*2010*2012 + 16*3*8*...*2010*2012 + ... + 4044121*3*8*...*2010)/(3*8*15*...*2010*2012)

a = (4*8*15*...*2010*2012 + 27*15*...*2010*2012 + 48*15*...*2010*2012 + ... + 12088263*15*...*2010*2012)/(3*8*15*...*2010*2012)

a = (4 + 27 + 48 + ... + 12088263)/(3*8*15*...*2010*2012)

Now we can find the sum of the series:

a = (1/2)(12088263)(12088264)/(3*8*15*...*2010*2012)

a = 730395726/2420835840

Therefore, the simplified expression for a is 730395726/2420835840.