Given the following information about an arithmetic sequence:
- \( t_n = 16 \) (The \( n \)-th term)
- \( d = 3 \) (The common difference)
- \( S_n = 51 \) (The sum of the first \( n \) terms)

We are to find \( t_1 \) (the first term) and \( n \) (the position of the term).

### Step 1: Use the formula for the \( n \)-th term of an arithmetic sequence.
The formula for the \( n \)-th term is:
\[
t_n = t_1 + (n - 1) \cdot d
\]
Given \( t_n = 16 \) and \( d = 3 \):
\[
16 = t_1 + (n - 1) \cdot 3
\]
\[
t_1 = 16 - 3(n - 1)
\]
\[
t_1 = 16 - 3n + 3 = 19 - 3n
\]

### Step 2: Use the formula for the sum of the first \( n \) terms.
The sum \( S_n \) of the first \( n \) terms is given by:
\[
S_n = \frac{n}{2} \cdot (t_1 + t_n)
\]
Substituting the given values \( S_n = 51 \) and \( t_n = 16 \):
\[
51 = \frac{n}{2} \cdot (t_1 + 16)
\]
Multiply both sides by 2:
\[
102 = n \cdot (t_1 + 16)
\]

### Step 3: Substitute \( t_1 = 19 - 3n \) into the sum equation.
\[
102 = n \cdot ((19 - 3n) + 16)
\]
\[
102 = n \cdot (35 - 3n)
\]
This simplifies to:
\[
102 = 35n - 3n^2
\]
Rearrange it into a standard quadratic equation:
\[
3n^2 - 35n + 102 = 0
\]

### Step 4: Solve the quadratic equation for \( n \).
Use the quadratic formula:
\[
n = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]
Here, \( a = 3 \), \( b = -35 \), and \( c = 102 \):
\[
n = \frac{35 \pm \sqrt{(-35)^2 - 4 \cdot 3 \cdot 102}}{6}
\]
\[
n = \frac{35 \pm \sqrt{1225 - 1224}}{6}
\]
\[
n = \frac{35 \pm \sqrt{1}}{6}
\]
\[
n = \frac{35 \pm 1}{6}
\]
This gives us two possible solutions:
\[
n = \frac{36}{6} = 6 \quad \text{or} \quad n = \frac{34}{6} = 5.67 \, \text{(not an integer, so discard this)}
\]
So, \( n = 6 \).

### Step 5: Find \( t_1 \).
Now, substitute \( n = 6 \) back into the equation for \( t_1 \):
\[
t_1 = 19 - 3n = 19 - 3(6) = 19 - 18 = 1
\]

### Final Answer:
- The first term \( t_1 = 1 \).
- The term \( n = 6 \).

Would you like further details on the process?

Here are five related questions you might consider:
1. What if the common difference \( d \) was negative?
2. How would the problem change if the sum \( S_n \) was different?
3. Can you find the general formula for the sum \( S_n \) in terms of \( t_1 \) and \( d \)?
4. How does changing \( t_n \) affect \( n \)?
5. What is the sum of the first 10 terms if \( t_1 \) and \( d \) are known?

**Tip:** When solving quadratic equations, always check if the solutions are integers, especially in contexts like sequences where \( n \) must be a whole number.

Learn how to find the first term and position in an arithmetic sequence with a detailed step-by-step solution. This problem covers key concepts in arithmetic sequences and quadratic equations, including the arithmetic sequence formula and quadratic equation formula. Suitable for students in Grades 9-12.

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