Sets

Sets

Question

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

a + (a + d) + (a + 2d) + ...........+ [a + (n - 1)d] = straight n over 2 left square bracket 2 straight a plus left parenthesis straight n minus 1 right parenthesis straight d right square bracket



Answer

Let P(n) : a + (a + d) + (a + 2d) + .............+ [a + (n - 1)d] = straight n over 2 left square bracket 2 straight a plus left parenthesis straight n minus 1 right parenthesis straight d right square bracket
I.     For n = 1,
      straight P left parenthesis 1 right parenthesis colon space straight a space equals space 1 half left square bracket 2 straight a space plus space left parenthesis 1 minus 1 right parenthesis straight d right square bracket
rightwards double arrow   straight a equals 1 half left square bracket 2 straight a space plus 0 right square bracket space rightwards double arrow space straight a space equals space 1 half left parenthesis 2 straight a right parenthesis space rightwards double arrow space straight a space equals space straight a

∴      P(1) is true
II.       Suppose the statement is true for n=m, straight m element of space straight N

∴       straight P left parenthesis straight m right parenthesis colon space straight a space plus space left parenthesis straight a space plus space straight d right parenthesis space plus space left parenthesis straight a space plus space 2 straight d right parenthesis space plus space left curly bracket straight a space plus space left parenthesis straight m space minus space 1 right parenthesis straight d right curly bracket space equals space straight m over 2 left square bracket 2 straight a space plus space left parenthesis straight m minus 1 right parenthesis straight d right square bracket .... (i)
III.   For n = m + 1,
       space space straight P left parenthesis straight m space plus space 1 right parenthesis space colon space straight a space plus space left parenthesis straight a space plus space straight d right parenthesis space plus space left parenthesis straight a space plus space 2 straight d right parenthesis space plus space....... space plus space left parenthesis straight a space plus space md right parenthesis space equals space fraction numerator straight m plus 1 over denominator 2 end fraction left square bracket 2 straight a space plus space left parenthesis straight m plus 1 minus 1 right parenthesis space straight d right square bracket
or      straight a plus left parenthesis straight a plus straight d right parenthesis plus left parenthesis straight a plus 2 straight d right parenthesis space plus space............ plus space left curly bracket straight a space plus space left parenthesis straight m minus 1 right parenthesis straight d right curly bracket space plus space left parenthesis straight a space plus space md right parenthesis space equals space fraction numerator straight m plus 1 over denominator 2 end fraction left square bracket 2 straight a space plus space md right square bracket
       From (i),
       straight a plus left parenthesis straight a plus straight d right parenthesis plus left parenthesis straight a plus 2 straight d right parenthesis plus.......... plus left curly bracket straight a plus left parenthesis straight m minus 1 right parenthesis straight d right curly bracket space equals space straight m over 2 left square bracket 2 straight a space plus space left parenthesis straight m space minus space 1 right parenthesis straight d right square bracket
∴      straight P left parenthesis straight m plus 1 right parenthesis space colon space straight m over 2 left square bracket 2 straight a space plus space left parenthesis straight m minus 1 right parenthesis space straight d right square bracket space plus space left parenthesis straight a space plus space md right parenthesis space equals space fraction numerator straight m plus 1 over denominator 2 end fraction left square bracket 2 straight a plus space md right square bracket
rightwards double arrow space space space space 1 half left square bracket 2 ma space plus space straight m squared straight d space minus space md space plus space 2 straight a space plus space 2 md right square bracket space equals space fraction numerator straight m space plus space 1 over denominator 2 end fraction left square bracket 2 straight a space plus space md right square bracket
rightwards double arrow space space space 1 half left square bracket 2 ma space plus space 2 straight a space plus space straight m squared straight d space plus space md right square bracket space equals space fraction numerator straight m plus space 1 space over denominator 2 end fraction left parenthesis 2 straight a space plus space md right parenthesis
space space rightwards double arrow space space 1 half left square bracket 2 straight a space left parenthesis straight m space plus space 1 right parenthesis space plus space md space left parenthesis space straight m plus space 1 right parenthesis right square bracket space equals space fraction numerator straight m plus 1 over denominator 2 end fraction left parenthesis 2 straight a space plus space md right parenthesis
rightwards double arrow space space space space space space fraction numerator straight m plus 1 over denominator 2 end fraction left square bracket 2 straight a space plus space md right square bracket space equals space fraction numerator straight m plus 1 over denominator 2 end fraction left square bracket 2 straight a space plus space md right square bracket
           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, P(n) is true for all straight n element of space straight N.



 

 



  

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