Sets

Sets

Question

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

1 plus 3 plus 5 plus........... space plus space left parenthesis 2 straight n minus 1 right parenthesis space equals space straight n squared

Answer

Let P(n): 1 plus 3 plus 5 plus............. plus left parenthesis 2 straight n minus 1 right parenthesis space equals space straight n squared
I.         For n = 1,
           P(1) : 1 = left parenthesis 1 right parenthesis squared rightwards double arrow space 1 space equals space 1
∴         P(1) is true
II.        Let the statement be true for n = m, straight m element of straight N
rightwards double arrow     P(m) : 1 + 3 + 5 + .................... + (2m - 1) = m2                    ...(i)
III.      For n = m + 1,
         P(m + 1) : 1 + 3 + 5 + .......... + [2 (m+1) - 1] = (m + 1)2
or      1 + 3 + 5 + ........... + (2m - 1) + (2m + 1) = (m + 1)2
        From (i),
         1 plus 3 plus 5 plus.............. space plus space left parenthesis 2 straight m space minus space 1 right parenthesis space equals space straight m squared

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rightwards double arrow    straight m squared space plus space 2 straight m space plus space 1 space equals space left parenthesis straight m space plus space 1 right parenthesis squared space rightwards double arrow space left parenthesis straight m space plus space 1 right parenthesis squared space equals space left parenthesis straight m space plus space 1 right parenthesis squared

∴       P (m + 1) is true.

∴       P(m) is true. rightwards double arrow space straight P left parenthesis straight m space plus space 1 right parenthesis is true.
         Hence, by the principal of mathematical induction, P(n) is true for all space space straight n element of straight N.


        

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