a.p.

let the number of terms be n

we know that sum of n terms of an A.P.  with a as first term, d as common diference is=

n/2[2a+(n-1)d]

here a=54, d= -3

so required sum

=n/2[108+(n-1)-3]

Thus we get an equation

n/2[111-3n]=513

So,

Solving we get,

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

n=18 or 19

Note the step where 3 has been cancelled thro'out the quadratic  equation. This is a very imp step as this helps reduce our calculation work and saves time also reduces the chance of making a calculation mistake.

Also note that in this case both answers will be valid as the 19 th tem comes out to be zero, so whether you take the sum of 18 terms or 19 terms the answer will be 513 .