1/1x3+1/2x4+1/3x5+… +1 / 2011x2013 + 1 / 2012x2014 = how much

1/1x3+1/2x4+1/3x5+… +1 / 2011x2013 + 1 / 2012x2014 = how much

Split term method: 1 / 1x3 + 1 / 2x4 + 1 / 3x5 + +1/2011x2013+1/2012x2014=1/2×（1-1/3+1/2-1/4+1/3-1/5+…… +1/2011-1/2013+1/2012-1/2014）=1/2×（1+1/2-1/2013-1/2014）=1/2×6077246/4054182=3038623/4054182

1x3 1 / 2 + 3x5 1 / 2 +. Up to 2011x2013 1 / 2 and

1 / 1x3 + 1 / 3x5 +. + 1 / 2011x2013 = 1 / 2x (1-1 / 3 + 1 / 3-1 / 5 +... + 1 / 2011-1 / 2013) = 1 / 2x (1-1 / 2013) = 1 / 2x2012 / 2013 = 1006 / 2013 help each other

Calculation process and answer of 1x3:2 + 3x5:2 + 5x7:2 + ·········· 31x33:2 + 33x35:2

The first term is equal to 1 / 1 minus 1 / 3, the second term is equal to 1 / 3 minus 1 / 5, the third term is equal to 1 / 5 minus 1 / 7, and so on. It is found that the subtracted (positive sign) of the latter term is equal to the subtracted (sign) of the former term, which can be eliminated by adding, that is, the negative third of the first term and the third of the second term are eliminated, The negative fifth of the second term and the fifth of the third term are eliminated, and so on. In the end, one of the first term and the negative thirty fifth of the last term are left. Finally, thirty four out of thirty-five are obtained
This method is often used in solving problems

1x3 / 1 + 3x5 / 1 + 5x7 / 1 +. + 1995x1997 / 1 + 1997x1999 / 1 =? Ingenious calculation

Original formula = 1 / 2x (1-1 / 3) + 1 / 2x (1 / 3-1 / 5) + 1 / 2x (1 / 5-1 / 7) + +1/2x(1/1995-1/1997)+1/2x(1/1997-1/1999)=1/2x(1-1/3+1/3-1/5+1/5-1/7+…… +1 / 1995-1 / 1997 + 1 / 1997-1 / 1999) = 1 / 2x (1-1 / 1999) = 1 / 2x 1998 / 1999 = 999 / 1999 (1999-9)