wiz-icon
MyQuestionIcon
MyQuestionIcon
1
You visited us 1 times! Enjoying our articles? Unlock Full Access!
Question

Using principle of mathematical induction, prove that n<11+12+13+...+1n for all natural numbers n2. [NCERT EXEMPLAR]

Open in App
Solution

Let Pn: n<11+12+13+...+1n for all natural numbers n2.Step I: For n=2,P2:LHS=21.414RHS=11+12=1+221+0.707=1.707As, LHS<RHSSo, it is true for n=2.Step II: For n=k,Let Pk: k<11+12+13+...+1k be true for some natural numbers n2.Step III: For n=k+1,Pk+1:LHS=k+1RHS=11+12+13+...+1k+1k+1>k+1k+1As, k+1>kkk+1<1kk+1<kk+1k+1-1k+1<kk+1-1k+1<kk+1k+1>k+1i.e. LHS<RHSSo, it is also true for n=k+1.Hence, Pn: n<11+12+13+...+1n for all natural numbers n2.

flag
Suggest Corrections
thumbs-up
0
Join BYJU'S Learning Program
similar_icon
Related Videos
thumbnail
lock
Mathematical Induction
MATHEMATICS
Watch in App
Join BYJU'S Learning Program
CrossIcon