1. **State the problem:** We have an arithmetic progression (A.P.) with first term $a_1 = 54$, common difference $d = 51 - 54 = -3$, and we want to find the number of terms $n$ such that the sum of these $n$ terms is 513. 2. **Formula for the sum of an A.P.:** The sum of the first $n$ terms of an A.P. is given by $$ S_n = \frac{n}{2} [2a_1 + (n-1)d] $$ 3. **Substitute known values:** $$ 513 = \frac{n}{2} [2 \times 54 + (n-1)(-3)] $$ 4. **Simplify inside the bracket:** $$ 513 = \frac{n}{2} [108 - 3(n-1)] = \frac{n}{2} [108 - 3n + 3] = \frac{n}{2} [111 - 3n] $$ 5. **Multiply both sides by 2:** $$ 1026 = n(111 - 3n) $$ 6. **Expand and rearrange into quadratic form:** $$ 1026 = 111n - 3n^2 $$ $$ 3n^2 - 111n + 1026 = 0 $$ 7. **Divide entire equation by 3:** $$ n^2 - 37n + 342 = 0 $$ 8. **Solve quadratic equation using the quadratic formula:** $$ n = \frac{37 \pm \sqrt{37^2 - 4 \times 1 \times 342}}{2} $$ Calculate discriminant: $$ 37^2 = 1369 $$ $$ 4 \times 342 = 1368 $$ $$ \sqrt{1369 - 1368} = \sqrt{1} = 1 $$ 9. **Find roots:** $$ n = \frac{37 \pm 1}{2} $$ So, $$ n_1 = \frac{37 + 1}{2} = 19 $$ $$ n_2 = \frac{37 - 1}{2} = 18 $$ 10. **Interpretation:** Number of terms must be a positive integer. Both 18 and 19 are positive integers, so either can be the number of terms for which the sum is 513. **Final answer:** The number of terms is $18$ or $19$.