Given series is:

54, 51, 48,...........

First term a = 54

Common difference d = 51 - 54 = -3

Let n is the number of terms.

Given, sum = 513

=> (n/2)*{2a + (n - 1)d} = 513

=> n{2a + (n - 1)d} = 513*2

=> n{2*54 + (n - 1)*(-3)} = 1026

=> n{108 - 3n + 3} = 1026

=> n{111 - 3n} = 1026

=> 111n - 3n² = 1026

=> 3n² - 111n + 1026 = 0

=> n² - 37n + 342 = 0            {divide by 3}

=> (n - 18)*(n - 19) = 0

=> n = 18, 19

Hence, number of terms is either 18 or 19