Solution:

a) Find the nth term of the sequence.

The given sequence is: $2, 10, 18, 26, 34, 42, 50, \ldots$.

1. Identify the difference between consecutive terms: $10 - 2 = 8, \; 18 - 10 = 8, \; 26 - 18 = 8, \; \text{and so on.}$ The sequence has a common difference of $d = 8$.

2. General form of an arithmetic sequence: The formula for the $n$-th term of an arithmetic sequence is: $a_n = a_1 + (n-1)d$ where $a_1$ is the first term and $d$ is the common difference.

3. Substitute $a_1 = 2$ and $d = 8$: $a_n = 2 + (n-1) \cdot 8$ Simplify: $a_n = 2 + 8n - 8 = 8n - 6$ So, the $n$-th term is: $a_n = 8n - 6$

b) Find the 200th term.

Using the formula $a_n = 8n - 6$: $a_{200} = 8 \cdot 200 - 6 = 1600 - 6 = 1594$ Thus, the 200th term is: $\boxed{1594}$

c) Which term of this sequence has the value 234?

We need to solve for $n$ when $a_n = 234$. Using the formula $a_n = 8n - 6$: $234 = 8n - 6$ Add 6 to both sides: $240 = 8n$ Divide by 8: $n = 30$ Thus, the 30th term of the sequence is: $\boxed{234}$

d) Show that 139 is not a term in the sequence.

For 139 to be a term, it must satisfy $a_n = 8n - 6$, where $n$ is an integer. Solve for $n$: $139 = 8n - 6$ Add 6 to both sides: $145 = 8n$ Divide by 8: $n = \frac{145}{8} = 18.125$ Since $n$ is not an integer, 139 is not a term in the sequence. Therefore: $\boxed{139 \text{ is not in the sequence.}}$

Summary of Results:

• a) $a_n = 8n - 6$
• b) $a_{200} = 1594$
• c) $234$ is the $30$-th term.
• d) $139$ is not a term in the sequence.

Let me know if you’d like further details or clarifications!

Related Questions:

1. What is the sum of the first 50 terms of this sequence?
2. How would the sequence change if the first term were $4$ and the difference were $10$?
3. Can you derive a formula for the sum of the first $n$ terms of this sequence?
4. What is the general method to check if a number belongs to an arithmetic sequence?
5. If the sequence were geometric, what would the formula for the $n$-th term look like?

Tip:

Always confirm that the value of $n$ is an integer when determining whether a number belongs to an arithmetic sequence.