How many terms of the progression 54 + 51 + 48 +... has the sum 513 ? Explain the double answer.

Explanation

To solve the problem of how many terms of the progression \(54 + 51 + 48 + \ldots\) have the sum of \(513\), we will follow these steps:

### Step 1: Identify the first term and common difference
The first term \(a\) of the arithmetic progression (AP) is:
\[ a = 54 \]
To find the common difference \(d\), we subtract the second term from the first term:
\[ d = 51 - 54 = -3 \]

### Step 2: Use the formula for the sum of the first \(n\) terms
The formula for the sum of the first \(n\) terms \(S_n\) of an arithmetic progression is:
\[ S_n = \frac{n}{2} \times (2a + (n - 1)d) \]
We know \(S_n = 513\), \(a = 54\), and \(d = -3\). Plugging in these values, we have:
\[ 513 = \frac{n}{2} \times (2 \times 54 + (n - 1)(-3)) \]

### Step 3: Simplify the equation
First, calculate \(2a\):
\[ 2a = 2 \times 54 = 108 \]
Now substitute this back into the equation:
\[ 513 = \frac{n}{2} \times (108 - 3(n - 1)) \]
This simplifies to:
\[ 513 = \frac{n}{2} \times (108 - 3n + 3) \]
\[ 513 = \frac{n}{2} \times (111 - 3n) \]

### Step 4: Eliminate the fraction
Multiply both sides by 2 to eliminate the fraction:
\[ 1026 = n(111 - 3n) \]
Expanding this gives:
\[ 1026 = 111n - 3n^2 \]
Rearranging the equation results in:
\[ 3n^2 - 111n + 1026 = 0 \]

### Step 5: Solve the quadratic equation
To solve the quadratic equation \(3n^2 - 111n + 1026 = 0\), we can use the quadratic formula:
\[ n = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Here, \(a = 3\), \(b = -111\), and \(c = 1026\). First, calculate the discriminant:
\[ b^2 - 4ac = (-111)^2 - 4 \times 3 \times 1026 \]
\[ = 12321 - 12312 = 9 \]
Now, substituting into the quadratic formula:
\[ n = \frac{111 \pm \sqrt{9}}{2 \times 3} \]
\[ n = \frac{111 \pm 3}{6} \]
Calculating the two possible values:
\[ n_1 = \frac{114}{6} = 19 \quad \text{and} \quad n_2 = \frac{108}{6} = 18 \]

### Step 6: Interpret the results
Thus, we have two possible values for \(n\):
\[ n = 18 \quad \text{or} \quad n = 19 \]

### Step 7: Verify the results
To understand why there are two answers, we can check the 18th and 19th terms:
- The 18th term \(T_{18}\):
\[ T_{18} = a + (18 - 1)d = 54 + 17(-3) = 54 - 51 = 3 \]
- The 19th term \(T_{19}\):
\[ T_{19} = a + (19 - 1)d = 54 + 18(-3) = 54 - 54 = 0 \]
The sum of the first 18 terms is \(513\), and the sum of the first 19 terms is also \(513\) because the 19th term is \(0\).

### Final Answer
Thus, the number of terms of the progression that sum to \(513\) is:
\[ \boxed{18 \text{ or } 19} \]