Using mathematical induction prove 2.3+3.4+4.5+..upto n terms = n (n^2+6n+11)/3.

Dear Student,
Please find below the solution to the asked query:

We have 2.3+3.4+4.5+...Clearly nth term will be n+1n+2Pn: 2.3+3.4+4.5+...+n+1n+2=nn2+6n+113Step 1: Base CaseFor n=12.3=11+6+113=1836=6Hence Pn is true for n=1Step 2: Inductive HypothesisLet Pn be true for n=k2.3+3.4+4.5+...+k+1k+2=kk2+6k+113 ;iStep 3: Inductive CaseConsider2.3+3.4+4.5+...+k+1k+2+k+1+1k+1+2=kk2+6k+113+k+2k+3 Using i=kk2+6k+113+k2+5k+6=k3+6k2+11k+3k2+15k+183=k3+6k2+11k+3k2+15k+1+11+63=k3+1+3k2+3k+6k2+12k+6+11k+113=k3+13+3k2+3k+6k2+2k+1+11k+13=k+13+6k+12+11k+13=k+1k+12+6k+1+1132.3+3.4+4.5+...+k+1k+2+k+1+1k+1+2=k+1k+12+6k+1+113Hence Pn is true for n=k+1Since Pn is true for n=1 and assumption of truth for n=k, gives us the resultthat Pn is also true for n=k+1.Hence by Principal of mathematical induction Pn is true for all nN

Hope this information will clear your doubts about this topic.

If you have any doubts just ask here on the ask and answer forum and our experts will try to help you out as soon as possible.

  • 12
What are you looking for?