Sets

Sets

Question

Prove the following by using the principle of mathematical induction for all straight n element of straight N colon

space space 1 plus fraction numerator 1 over denominator 1 plus 2 end fraction plus fraction numerator 1 over denominator 1 plus 2 plus 3 end fraction plus....... plus fraction numerator 1 over denominator 1 plus 2 plus 3 plus........ plus straight n end fraction space equals space fraction numerator 2 straight n over denominator straight n plus 1 end fraction

Answer

Let straight P left parenthesis straight n right parenthesis colon space space space 1 plus fraction numerator 1 over denominator 1 plus 2 end fraction plus fraction numerator 1 over denominator 1 plus 2 plus 3 end fraction plus...... plus fraction numerator 1 over denominator 1 plus 2 plus 3 plus....... plus straight n end fraction space equals space fraction numerator 2 straight n over denominator straight n plus 1 end fraction
I.      For n = 1,
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∴      P(1) is true
II.      Let the statement be true for n = m,   straight m element of space straight N
       P(m) : 1 plus fraction numerator 1 over denominator 1 plus 2 end fraction plus fraction numerator 1 over denominator 1 plus 2 plus 3 end fraction plus..... plus fraction numerator 1 over denominator 1 plus 2 plus 3 plus......... plus straight m end fraction space equals space fraction numerator 2 straight m over denominator straight m plus 1 end fraction     ...(i)
III.   For n = m + 1,
       straight P left parenthesis straight m plus 1 right parenthesis colon space 1 space plus space fraction numerator 1 over denominator 1 plus 2 end fraction space plus space fraction numerator 1 over denominator 1 plus 2 plus 3 end fraction plus space.......... plus space fraction numerator 1 over denominator 1 plus 2 plus 3 plus........ plus left parenthesis straight m plus 1 right parenthesis end fraction space equals space fraction numerator 2 left parenthesis straight m plus 1 right parenthesis over denominator straight m plus 1 plus 1 end fraction   
 
or 1 plus fraction numerator 1 over denominator 1 plus 2 end fraction plus fraction numerator 1 over denominator 1 plus 2 plus 3 end fraction plus...... plus space fraction numerator 1 over denominator 1 plus 2 plus 3 plus....... plus straight m end fraction plus fraction numerator 1 over denominator 1 plus 2 plus 3 plus....... plus left parenthesis straight m plus 1 right parenthesis end fraction space equals space fraction numerator 2 straight m plus 2 over denominator straight m plus 2 end fraction
    From (i),
      1 plus fraction numerator 1 over denominator 1 plus 2 end fraction plus fraction numerator 1 over denominator 1 plus 2 plus 3 end fraction plus.......... plus fraction numerator 1 over denominator 1 plus 2 plus 3 plus.......... plus straight m end fraction space equals space fraction numerator 2 straight m over denominator straight m plus 1 end fraction 
      Also,  1, 2 , 3,..........., m+1 are in A.P.
rightwards double arrow  1 plus 2 plus 3 plus........... plus left parenthesis straight m plus 1 right parenthesis equals fraction numerator straight m plus 1 over denominator 2 end fraction open square brackets 1 plus space left parenthesis straight m plus 1 right parenthesis close square brackets space equals space fraction numerator left parenthesis straight m plus 1 right parenthesis left parenthesis straight m plus 2 right parenthesis over denominator 2 end fraction
                                                                          
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∴     space space straight P left parenthesis straight m plus 1 right parenthesis space colon space fraction numerator 2 straight m over denominator straight m plus 1 end fraction plus space fraction numerator 1 over denominator begin display style fraction numerator straight m plus 1 over denominator 2 end fraction left parenthesis straight m plus 2 right parenthesis end style end fraction space equals space fraction numerator 2 straight m space plus 2 space over denominator straight m space plus space 2 end fraction space rightwards double arrow space fraction numerator 2 straight m over denominator straight m plus 1 end fraction plus fraction numerator 2 over denominator left parenthesis straight m plus 1 right parenthesis left parenthesis straight m plus 2 right parenthesis end fraction space equals space fraction numerator 2 straight m plus 2 over denominator straight m plus 2 end fraction
rightwards double arrowfraction numerator 2 over denominator straight m plus 1 end fraction open square brackets straight m plus fraction numerator 1 over denominator straight m plus 2 end fraction close square brackets space equals space fraction numerator 2 straight m plus 2 over denominator straight m plus 2 end fraction space rightwards double arrow space fraction numerator 2 over denominator straight m plus 1 end fraction open square brackets fraction numerator straight m squared plus 2 straight m plus 1 over denominator straight m plus 2 end fraction close square brackets space equals space fraction numerator 2 straight m space plus space 2 over denominator straight m plus 2 end fraction
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         which is true.
∴      P(m + 1) is true.

∴      P(m) is true rightwards double arrow P(m + 1) is true.
Hence, by the principle of mathematical induction, the statement P(n) is true for all straight n element of space straight N.


       


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