Question

Q=\(\dfrac{3}{1\cdot3}+\dfrac{3}{3\cdot5}+.....+\dfrac{3}{101\cdot103}+\dfrac{3}{103\cdot105}\)

Answer 1

Q = 3 .(\(\dfrac{1}{1.3}\) + \(\dfrac{1}{3.5}\) + ...+\(\dfrac{1}{101.103}\) + \(\dfrac{1}{103.105}\) )

Q = 3. (1/1-1/3+1/3-1/5+1/5-...+1/103-1/105)

Q = 3. (1/1-1/105)

Q = 3. 104/105

Q = 104/35

Answer 2

\(\dfrac{3}{2}\)Q=\(\dfrac{2}{1.3}+\dfrac{2}{3.5}+\dfrac{2}{5.7}\)\(+...+\dfrac{2}{101.103}+\dfrac{2}{103.105}\)

=\(\dfrac{1}{1}-\dfrac{1}{3}+\dfrac{1}{3}-\dfrac{1}{5}+...+\dfrac{1}{101}-\dfrac{1}{103}+\dfrac{1}{103}-\dfrac{1}{105}\)

=\(\dfrac{1}{1}-\dfrac{1}{105}\)

\(\Rightarrow\)\(\dfrac{3}{2}\)Q=\(\dfrac{104}{105}\)

\(\Rightarrow\)Q=\(\dfrac{104}{105}:\dfrac{3}{2}\)

\(\Rightarrow\)Q=\(\dfrac{208}{315}\)

Answer 3

Q= \(\dfrac{3}{2}.\left(\dfrac{1}{1}-\dfrac{1}{3}+\dfrac{1}{3}-\dfrac{1}{5}+...+\dfrac{1}{101}-\dfrac{1}{103}+\dfrac{1}{103}-\dfrac{1}{105}\right)\)

Q= \(\dfrac{3}{2}.\left(\dfrac{1}{1}-\dfrac{1}{105}\right)\)

Q= \(\dfrac{3}{2}.\dfrac{104}{105}\)

Q=\(\dfrac{52}{35}\)