`S = 1/( 5 xx 9 ) + 1/( 9xx13 ) +.....+1/( 41xx45)`
`= 1/4 xx ( 1/( 5 xx 9 ) + 1/( 9xx13 ) +.....+1/( 41xx45))`
`= 1/4 xx ( 1/5 - 1/9 + 1/9 - 1/13 +....+1/41-1/45)`
`= 1/4 xx ( 1/5 - 1/45 )`
`= 2/45`
`4S = 1/(5xx9) + 1/(9xx13) + ... + 4/(41xx45)`
`4S = 1/5 - 1/9 + 1/9 - 1/13 + ... + 1/41 - 1/45`
`4S = 8/45`
`=> S = 2/45`
4S = \( (\dfrac{1}{5x9})\)+\( (\dfrac{1}{9x13})\)+\( (\dfrac{1}{13x17})\)+....\( (\dfrac{1}{41x45})\)
4S = [\( (\dfrac{1}{5x9})+(\dfrac{1}{9x13})+(\dfrac{1}{13x17})+...+(\dfrac{1}{41x45})\)]
4S = \((\dfrac{1}{5}-\dfrac{1}{9}+\dfrac{1}{ 9}-\dfrac{1}{ 13}+...+\dfrac{1}{ 41}-\dfrac{1}{ 45}\))
4S = (\(\dfrac{1}{ 5}-\dfrac{1}{ 45}\))
4S = \(\dfrac{8}{ 45}\)
S = \(\dfrac{2}{45}\)
S = \(\dfrac{32}{ 45}\)