how many terms of the AP 72,69,66,..make the sum 897 ?

 Suppose we have added p no. of terms to get sum 897

WE have S_p =897

                 a =72

                 d =69 -72 =-3

We know that S_n =n/2 *(2a + (n-1)d)

                     S_p = p/2 *(2 *72 + (p-1) *(-3))

               897 = p/2(144 -3p +3)

           897 = p/2 (147 -3p)

       897 *2  =147p -3p²

       1794  = 147p - 3p²

  3p² -147 p +1794 =0

   3(p² -49p +598)= 0

p² -49p +598 =0

p² -26p -23p +598 =0

p(p-26)-23(p- 26)=0

(p-26)(p-23)=0

p=26 or p=23

thus S₂₆ =897 or S₂₃ =897

From above Sum of 23 terms are equal to sum of 26 terms this show that 24, 25, 26.... are Zero

Therefore Sum of 23 terms are 897