Express the following numbers as sum of two consecutive integers

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70.

Express the following numbers as sum of two consecutive integers,

(i) (23)2 (ii) (17)2 (iii) (27)2


(i) We have n = 23

space space space space space space space space space space space space space space space space space fraction numerator n squared minus 1 over denominator 2 end fraction equals fraction numerator 23 squared minus 1 over denominator 2 end fraction equals fraction numerator 529 minus 1 over denominator 2 end fraction

        = space space space space space space space 528 over 2 equals 264

space space space space space space space space space space space space space fraction numerator straight n squared plus 1 over denominator 2 end fraction equals fraction numerator 23 squared plus 1 over denominator 2 end fraction equals 530 over 2 equals 265

       space space space space space n squared equals fraction numerator n squared minus 1 over denominator 2 end fraction plus fraction numerator n squared plus 1 over denominator 2 end fraction

space space rightwards double arrow   232 = 264 + 265


(ii)    We have n = 17

  space space space space space space space space space space space space space space space space fraction numerator straight n squared minus 1 over denominator 2 end fraction equals fraction numerator 17 squared minus 1 over denominator 2 end fraction equals fraction numerator 289 minus 1 over denominator 2 end fraction

         =space space space space space space space space 288 over 2 equals 144


space space space space space space space space space space space space fraction numerator straight n squared plus 1 over denominator 2 end fraction equals fraction numerator 17 squared plus 1 over denominator 2 end fraction equals fraction numerator 289 plus 1 over denominator 2 end fraction equals 145

    172 = 144 + 145


(iii)   We have n = 27

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           space space space space space space space space space fraction numerator n squared plus 1 over denominator 2 end fraction equals fraction numerator 27 squared plus 1 over denominator 2 end fraction equals 365

        (27)2 = 364 + 365



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