Question

A=1/15+1/35+1/63+1/99+...+1/9999

Answer 1

A = 1/3.5 + 1/5.7 + 1/7.9 + 1/9.11 + ... + 1/99.101

= 1/2.(2/3.5 + 2/5.7 + 2/7.9 + ... + 2/99.101)

= 1/2.(1/3 - 1/5 + 1/5 - 1/7 + 1/7 - 1/9 + ... + 1/99 - 1/101)

= 1/2.(1/3 - 1/101)

= 1/2.98/303

= 49/303

Answer 2

A = 1/15 + 1/35 + 1/ 63 + 1/99 + ...+ 1/9999

A     = 1/(3x5) + 1/(5x7) + 1/(7x9) + 1/(9x11) + ... + 1/(99 x 101)

Ax2 = 2/(3x5) + 2/(5x7) + 2/(7x9) + 2/(9x11) + ... + 2/(99 x 101)

Ax2 = 1/3 – 1/5 + 1/5 – 1/7 + 1/7 – 1/9 + 1/9 – 1/11 + ...+ 1/99 – 1/101

Ax2 = 1/3 – 1/101 = 98/303

A    =  98/303 : 2

A    =  49/303