If `1+ab+a^(2)b^(2)+a^(3)b^(3)+"..."+infty =(xy)/(x+y-1)` are the sum of infinire geometric series whose first terms are `1,2,3,"….",p` and whose common ratios are `S_(1),S_(2),S_(3),"....,"S_(p)``(1)/(2),(1)/(3),(1)/(4),"..."(1)/(p+1)` respectively, prove that `S_(1)+S_(2)+S_(3)+"....+"S_(p)=(p(p+3))/(2)`.

Explanation

` therefore S_(p)=(p)/(1-(1)/(p+1))=(p+1)`
` therefore S_(1)=2,S_(2)=3,S_(3)=4,"...."`
` therefore LHS" " S_(1)+S_(2)+S_(3)+"....+"S_(p)`
`=1+2+3+4+"....+"(p+1)=(p)/(2)(2+p+1)`
`=(p(p+3))/(2)=RHS`