The problem states that $2x$, $4x + 1$, and $14$ are the first three terms of an arithmetic sequence. The task is to find the 10th term of this sequence.

Step-by-Step Solution:

Step 1: Understanding Arithmetic Sequence

For an arithmetic sequence, the difference between consecutive terms is constant: $\text{Second term} - \text{First term} = \text{Third term} - \text{Second term}.$

Step 2: Set up the equation

Given: $2x, \, 4x + 1, \, 14$ From the property of an arithmetic sequence: $(4x + 1) - (2x) = 14 - (4x + 1).$

Step 3: Simplify the equation

1. Simplify the left-hand side: $(4x + 1) - 2x = 2x + 1.$
2. Simplify the right-hand side: $14 - (4x + 1) = 14 - 4x - 1 = 13 - 4x.$ Equating both sides: $2x + 1 = 13 - 4x.$

Step 4: Solve for $x$

$2x + 4x = 13 - 1,$ $6x = 12,$ $x = 2.$

Step 5: Find the terms of the sequence

Substitute $x = 2$ into the terms:

1. First term: $2x = 2(2) = 4$,
2. Second term: $4x + 1 = 4(2) + 1 = 9$,
3. Third term: $14$ (given).

Thus, the arithmetic sequence is: $4, 9, 14, \dots$

Step 6: General formula for the $n$-th term

The general formula for an arithmetic sequence is: $a_n = a_1 + (n-1)d,$ where:

• $a_1 = 4$ (first term),
• $d = 9 - 4 = 5$ (common difference).

For the 10th term ($n = 10$): $a_{10} = 4 + (10-1)(5),$ $a_{10} = 4 + 9(5),$ $a_{10} = 4 + 45 = 49.$

Final Answer:

The 10th term is $\mathbf{49}$.
The correct choice is C.

Do you have any questions or would you like additional details?
Here are 5 related questions to explore:

1. What is the formula for the sum of the first $n$ terms of an arithmetic sequence?
2. How can you identify an arithmetic sequence from a list of numbers?
3. What happens if the sequence given is geometric instead of arithmetic?
4. Can you solve for the 20th term using the same method?
5. How do you calculate the difference ($d$) if the sequence is missing terms?

Tip: Always verify the common difference when working with arithmetic sequences to ensure consistency.