Question

if the sum of first m terms of an AP is amsquare+bm find its common difference

Answer 1

Solution :

     Let the common difference of AP= x
       and first term of AP = a
   We know that
               Sum of m terms = m² + bm  .....(Given)-------(Ist)
   Also ,
     Sum of m terms = (first term +last term )*Number of terms /2
           here , first term = a and last term = (m-1)d
Sum of m terms = (a +(m-1)d)*m /2 ----- .....(2nd)
From equation Ist and 2nd , we get
                         m² + bm =(a +(m-1)d)*m /2
                       or m (m+b) = (a+(m-1)d)*m/2
                       or m+b = (a +(m-1)d)/2
                     or 2 (m+b) = a +(m-1)d
                     or 2(m+b) -a = (m-1)d
                    or (2(m+b) -  a )/ (m-1) =d
                   or d = (2(m+b) -  a )/ (m-1)

  Hence Common difference (d) is   (2(m+b) -  a )/ (m-1)          

  Hope this helps You!!
     Thank You !!