Question

how many terms of the AP 54, 51, 48, 45,.... be is taken so that their sum is 513

Answer 1

here a=54, d = 51-54=-3
Sn=513
We know   Sn=n/2(2a+(n-1)d)
                  513=n/2(2.54+(n-1)-3)
                    513x2=n(108-3n+3)
                    1026=108n-3n^2+3n
                    3n^2-111n+1026=0
take 3 as common we get
                         n^2-37n+342=0
              by factorisation
                         n^2-19n-18n+342=0
                          n(n-19)-18(n-19)=0
                              (n-19)(n-18)=0
               either(n-19)=0 or (n-18)=0
               either n=19 or n=18
      we get
                     n=18 or n= 19