This is a problem related to an arithmetic sequence.

Given:

• $U_5 = 4$
• $U_{25} = 14$

We need to find $U_{21}$.

The formula for the $n$-th term of an arithmetic sequence is:

$U_n = a + (n - 1) d$ where:

• $a$ is the first term of the sequence,
• $d$ is the common difference between consecutive terms.

From the information given, we can set up the following two equations:

1. $U_5 = a + (5 - 1) d = a + 4d = 4$
2. $U_{25} = a + (25 - 1) d = a + 24d = 14$

Now, solve these equations step by step.

Step 1: Subtract the first equation from the second:

$(a + 24d) - (a + 4d) = 14 - 4$ $20d = 10$ $d = \frac{10}{20} = 0.5$

Step 2: Substitute $d = 0.5$ back into the first equation to find $a$:

$a + 4(0.5) = 4$ $a + 2 = 4$ $a = 2$

Step 3: Use the values of $a = 2$ and $d = 0.5$ to find $U_{21}$:

$U_{21} = a + (21 - 1) d = 2 + 20(0.5) = 2 + 10 = 12$

So, the value of $U_{21}$ is 12 (Answer: A).

Do you have any further questions or would you like more details?

Here are 5 related questions to explore further:

1. What is the general form of an arithmetic sequence?
2. How would you find the sum of the first 25 terms in this arithmetic sequence?
3. If $U_5 = 4$ and $U_{25} = 14$, what is the formula for the $n$-th term of this sequence?
4. How does the common difference affect the progression of the sequence?
5. Can you derive the first term of the sequence if you know any two terms and the common difference?

Tip: In arithmetic sequences, once you know the common difference $d$, finding any term in the sequence becomes a straightforward application of the formula.