The sum $1/(1\times 2)+1/(2\times 3)+1/(3\times 4)$ + … + 1/$(20\times 21)$ equals:

Split each term: 1/(k(k + 1)) = 1/k − 1/(k + 1). Adding, everything cancels in pairs and only the first and last pieces survive: 1 − 1/21 = 20/21. (1/21 is the piece that is subtracted, not the sum; 19/20 stops one term early at 1 − 1/20; 21/20 is the reciprocal of the answer.) The first two terms already give 1/2 + 1/6 = 2/3 = 1 − 1/3, which is the pattern.