` A = ` ` 1/(11 xx 13) ` $+$ ` 1/(13 xx 15) ` $+$ ` 1/(15 xx 17) ` $+$ ` 1/(17 xx 19) ` $+...+$ ` 1/(97 xx 99) `
$=$ ` 1/2 xx 2 ` ` xx ` ` (1/(11 xx 13) ` $+$ ` 1/(13 xx 15) ` $+$ ` 1/(15 xx 17) ` + ` 1/(17 xx 19) ` $+...+$ ` 1/(97 xx 99)) `
$=$ ` 1/2 xx ` ` (2/(11 xx 13) ` $+$ ` 2/(13 xx 15) ` $+$ ` 2/(15 xx 17) ` $+$ ` 2/(17 xx 19) ` $+...+$ ` 2/(97 xx 99)) `
` = 1/2 xx (1/11 - 1/13 + 1/13 - 1/15 + 1/15 - 1/17 + 1/17 - 1/19 +...+ 1/97 - 1/99) `
$=$ ` 1/2` ` xx ` ` (1/11 - 1/99 ) `
$=$ ` 1/2 ` ` xx ` ` (9/99 - 1/99 ) `
$=$ ` 1/2 ` ` xx ` ` 8/99 `
$=$ ` 4/99 `
