Question

1/(1×2)+1/(2×3)+...+n(n+1)=n/n+1

Answer 1

Answer:

We have;

LHS = 1/1•2 + 1/2•3 + 1/3•4 + 1/4•5 + 1/5•6+................+ 1/n(n+1)

= (2-1)/1•2 + (3-2)/2•3 + (4-3)/3•4 +

(5-4)/4•5 + (6-5)/5•4 +............

............+ {(n+1)-n}/n•(n+1)

= (1 - 1/2) + (1/2 - 1/3) + (1/3 - 1/4) +

(1/4 - 1/5) + (1/5 - 1/6) + ..........

...........+ {1/n - 1/(n+1)}

= 1 - 1/(n+1)

= (n+1-1)/(n+1)

= n/(n+1)

= RHS

Hence proved.