1/3.6+1/6.9+1/9.12+..+1/3n(3n+3)=n/9(n+1)

1.	1/3.6+1/6.9+1/9.12+..+1/3n(3n+3)=n/9(n+1)
Answer» Step 1 put (n=1)L.H.S.1/3.6=1/18R.H.S.n/9(n+1)=1/9(1+1)=1/9.2=1/18Step 2 Let it be true for (n=k)1/3.6+1/6.9+1/9.12+...+1/3k(3k+3)=k/9(k+1)Step 3 Put (n=k+1)1/3.6+1/6.9+1/9.12+...+1/3(k+1)(3(k+1)+3)=k+1/9(k+1+1)=1/3.6+1/6.9+1/9.12+...+1/3k(3k+3)+1/(3k+3)(3k+6)=k+1/3(3k+6)L.H.S.=k/3(3k+3)+1/(3k+3)(3k+6)=1/3k+3(k/3+1/3k+6)=1/3k+3(3k^2+6k+3/3(3k+6))=1/3k+3((k+1)(3k+3)/3(3k+6))=1(k+1/3(3k+6))=k+1/3(3k+6)=R.H.S.

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