Given Information:
The series is given as 54, 51, 48, ...

Objective:
To determine how many terms of the series should be taken so that their sum is 513.

Approach:
1. We need to identify the pattern in the given series.
2. Subtracting 3 from each term in the series, we get the series 51, 48, 45, ...
3. We observe that each term in the new series is obtained by subtracting 3 from the previous term.
4. This indicates that the given series is an arithmetic series with a common difference of -3.

Formula for the sum of an arithmetic series:
The sum of an arithmetic series is given by the formula:
Sn = n/2 * (2a + (n-1)d)
where Sn is the sum of the series, n is the number of terms, a is the first term, and d is the common difference.

Calculation:
Let's substitute the given values into the formula and solve for n.
Sn = 513 (given sum)
a = 54 (first term)
d = -3 (common difference)

Using the formula, we have:
513 = n/2 * (2*54 + (n-1)(-3))

Simplifying the equation:
513 = n/2 * (108 - 3n + 3)
513 = n/2 * (111 - 3n)

Multiplying both sides by 2 to eliminate the fraction:
1026 = n * (111 - 3n)

Rearranging the equation to form a quadratic equation:
3n^2 - 111n + 1026 = 0

Solving the quadratic equation using factorization or the quadratic formula, we find two solutions for n:
n = 6 and n = 57

Explanation of the Double Answer:
The double answer is obtained because the given arithmetic series is decreasing. When we sum the first 6 terms of the series, we get a sum of 513. However, if we continue to add more terms, the sum will decrease. Hence, the sum of the first 57 terms of the series is also 513.

Conclusion:
To obtain a sum of 513, we can take either 6 terms or 57 terms of the series, depending on the context and the desired level of precision.