Prove by induction:

1/1.2 + 1/2.3 + 1/3.4 + ..... to n terms = n/(n+1)

if the question is prove that p(n)= 1/1.2+ 1/2.3+ 1/3.4+....+ 1/n(n+1)= n/n+1

then for n=1 p(n)= 1/2= LHS RHS=1/1+1 =1/2

LHS=RHS

assume p(n) is true for n=k

1/1.2+  1/2.3+...+ 1/k.(k+1) = k/k+1

verify p(n) is true for n=k+1

= 1/1.2+  1/2.3  +1/3.4+...+ 1/k(k+1)+1/(k+1)(k+2)

=  [1/1.2+1/2.3+...+1/k(k+1)]  + 1/(k+1)(k+2)

=k/k+1 +1/ (k+1)(k+2)

by taking LCM

k(k+2)+1/ (k+1)(k+2)

k2+2k+1/ (k+1)(k+2)

k2+k+k+1/ (k+1)(k+2)  on simplification

(k+1)(k+1)/ (k+1)(k+2)

=(k+1)/ (k+2)

since k+1=n (assumed above)

=n/n+1

therefore by principle of mathematical induction the given statement is true for all n belongs to N

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 for n=1 p(n)= 1/2= LHS RHS=1/1+1 =1/2

LHS=RHS

assume p(n) is true for n=k

1/1.2+  1/2.3+...+ 1/k.(k+1) = k/k+1

verify p(n) is true for n=k+1

= 1/1.2+  1/2.3  +1/3.4+...+ 1/k(k+1)+1/(k+1)(k+2)

=  [1/1.2+1/2.3+...+1/k(k+1)]  + 1/(k+1)(k+2)

=k/k+1 +1/ (k+1)(k+2)

by taking LCM

k(k+2)+1/ (k+1)(k+2)

k2+2k+1/ (k+1)(k+2)

k2+k+k+1/ (k+1)(k+2)  on simplification

(k+1)(k+1)/ (k+1)(k+2)

=(k+1)/ (k+2)

since k+1=n (assumed above)

=n/n+1

therefore by principle of mathematical induction the given statement is true for all n belongs to N

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