[" 52."[(1)/(1times2)+(1)/(2times3)+(1)/(3times4)+...+(1)/(99times100)]=?],[[" (a) "(1)/(9900)," (b) "(99)/(100)," (c) "(100)/(99)," (d) "(1000)/(99)]]

[(1)/(1*2)+(1)/(2*3)+(1)/(3*4)+....+(1)/(99*100)] =?

The value of (1)/(1xx2)+(1)/(2xx3)+(1)/(3xx4)+..... +(1)/(99xx100) is

[(1)/(1xx2)+(1)/(2xx3)+(1)/(3xx4)+,+(1)/(99xx100)]=

(1)/(1+sqrt(2))+(1)/(sqrt(2)+sqrt(3))+(1)/(sqrt(3)+sqrt(4))+...(1)/(sqrt(99)+sqrt(100))

The following steps are involved in finding the value of 100 (3)/(4) xx 99(1)/(4) by using appropriate identify . Arrange them in sequential order . (A) (100 + (3)/(4)) (100 - (3)/(4)) = (100)^(2) - ((3)/(4))^(2) [ because (a + b) (a - b) = a^(2) - b^(2)] (B) 10000 - (9)/(16) = 9999 + 1 - (9)/(16) = 9999 (7)/(16) (C) (100)^(2) - ((3)/(4))^(2) = 10000 - (9)/(16) (D) 100 (3)/(4) xx 99 (1)/(4) = (100 + (3)/(4)) (100 - (3)/(4))

The sum of the first 99terms of the series (3)/(4)+(5)/(36)+(7)/(144)+(9)/(400)+backslash*(99)/(100) (b) (999)/(1000) (c) (9999)/(10000) (d) 1

The following steps are involved in finding the value of 10 (1)/(3) xx 9(2)/(3) by using an appropriate indentity . Arrange them in sequential order . (A) (10)^(2) - ((1)/(3))^(2) = 100 - (1)/(9) (B) 10(1)/(3) xx 9(2)/(3) = (10 + (1)/(3)) (10 - (1)/(3)) (C) (10 + (1)/(3)) (10 - (1)/(3)) = (10)^(2) - ((1)/(3))^(2) [because (a + b) (a -b) = (a^(2) - b^(2))] (D) 100 - (1)/(9) = 99 + 1 - (1)/(9) = 99(8)/(9)