In an arithmetic series, the general form of the $n$-th term can be expressed as:

$a_n = a_1 + (n-1) \cdot d$ where $a_1$ is the first term and $d$ is the common difference between consecutive terms.

Let us define the following terms for the problem:

• $a_p = 2057$ (the $p$-th term),
• $a_{p+5} = 2102$ (the $(p+5)$-th term).

Step 1: Express both terms using the formula

From the general formula for the $n$-th term, we can write the two given terms as:

a_p = a_1 + (p-1) \cdot d = 2057 \tag{1}

a_{p+5} = a_1 + (p+5-1) \cdot d = a_1 + (p+4) \cdot d = 2102 \tag{2}

Step 2: Subtract equation (1) from equation (2)

This will eliminate $a_1$ and give us an equation involving only $d$:

[ [a_1 + (p+4) \cdot d] - [a_1 + (p-1) \cdot d] = 2102 - 2057 ]

Simplifying:

$(p+4) \cdot d - (p-1) \cdot d = 45$

$[(p+4) - (p-1)] \cdot d = 45$

$(5) \cdot d = 45$

Thus, the common difference is:

$d = \frac{45}{5} = 9$

Step 3: Find the value of $a_1$

Now that we know $d = 9$, substitute this value back into equation (1) to find $a_1$:

$a_1 + (p-1) \cdot 9 = 2057$

$a_1 + 9(p-1) = 2057$

Solving for $a_1$:

$a_1 = 2057 - 9(p-1)$

Step 4: Find the $(p+8)$-th term

Now we can use the formula for the general term to find the $(p+8)$-th term:

$a_{p+8} = a_1 + (p+8-1) \cdot d = a_1 + (p+7) \cdot 9$

Substitute the value of $a_1$ from the previous equation:

$a_{p+8} = [2057 - 9(p-1)] + (p+7) \cdot 9$

Simplifying:

$a_{p+8} = 2057 - 9(p-1) + 9(p+7)$

$a_{p+8} = 2057 - 9(p-1) + 9p + 63$

$a_{p+8} = 2057 + 63 - 9(p-1) + 9p$

$a_{p+8} = 2120$

Thus, the $(p+8)$-th term is:

$\boxed{2120}$

Would you like further clarification or a breakdown of any part of this process? Here are some related questions:

1. How can we solve for the first term of an arithmetic series if we are given two specific terms?
2. What would change in the solution if the common difference was negative?
3. How do we express the $n$-th term of an arithmetic series and how can it be used to find specific terms?
4. How can we use the formula for an arithmetic sequence to find the sum of the first $n$ terms?
5. How do you verify the common difference from two given terms in an arithmetic series?

Tip: In problems involving arithmetic sequences, focusing on eliminating variables (like $a_1$) through subtraction often simplifies the solution process.