Question

calculati;
(1/2*3+1/3*4+........+1/31*32)-5/32
1/2011+(1/1*2+1/2*3+1/3*4+......+1/2010*201)
1/10*11+1/11*12+112*13+.....+1*50)-(2,39-1,59)
(2/1*3+2/3*5+2/5*7+.....+2/79*81)-(1/1*2+1/2*3+1/3*4+.....+1/80*81)
(3/1*4+3/4*7+3/7*10+......+3/52*53)+(1/10*11+1/11*12+/12*13+.....+1/54*55)
VA ROG,AJUTATI-MA

Answer 1

A = 1/2·3 +1/3·4 +1/4·5 +......+1/31·32 = 1/2 -1/3+1/3-1/4+1/4-1/5+ + .......+1/31 - 1/32 = 1/2 -1/32⇒A-5/32 = 1/2 - 6/32 =1/2-3/16=5/16 B=1/1·2 +1/2·3 +1/3·4 +.......+1/2010·2011 = 1/1 - 1/2 +1/2 -1/3 + 1/3 - 1/4 +..........+1/2010 - 1/2011 = 1/1 - 1/2011 ⇒ 1/2011 - B = 1/2011+ 1 - 1/2011 = 1

C= 1/10·11 + 1/11·12 +.....+ 1/50·51 = 1/10 - 1/11 + 1/11 - 1/12 +.....+1/50 - 1/51 = 1/10 - 1/51 = 41/510    2,39 -1,59 = 0,8 =4/5  ⇒ C- 4/5 = 41/510 - 4/5 = - 1999/510

D = 1/1 -1/3 + 1/3-1/5 +.......+ 1/79 -1/81 = 1- 1/81 = 80/81   E=1/1·2 + 1/2·3 + 1/3·4 +......1/80·81 = 1- 1/81 =  80/81 ⇒  D-E=0

F = 3/1·4 + 3/4·7 +......+3/52·55 = 1/1-1/4 +1/4 -1/7+..1/52 -1/55 = 1-1/55 = 54/55    G=1/10·11 +1/11·12 +....+1/54·55 =  1/10-1/55 =9/110 ⇒ F+G = 54/55 + 9/110 = 117/110