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.