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This problem asks to find the arithmetic mean between two numbers from a sequence. The numbers given are \( -1 \) and \( \frac{1}{4} \), and it is part of an arithmetic sequence.

The formula for the arithmetic mean between two terms \( a \) and \( c \) is:

\[
b = \frac{a + c}{2}
\]

Here, \( a = -1 \) and \( c = \frac{1}{4} \).

Let's calculate:

\[
b = \frac{-1 + \frac{1}{4}}{2}
\]

First, simplify the numerator:

\[
-1 + \frac{1}{4} = \frac{-4}{4} + \frac{1}{4} = \frac{-3}{4}
\]

Now, divide by 2:

\[
b = \frac{-3}{4} \div 2 = \frac{-3}{4} \times \frac{1}{2} = \frac{-3}{8}
\]

So, the arithmetic mean is \( \frac{-3}{8} \).

Next, we need to check which option corresponds to \( \frac{-3}{8} \). The available answers are:

1. 57
2. 41
3. 14
4. 3

Since the result \( \frac{-3}{8} \) doesn't directly match these options, it's likely the question involves interpreting the mean as an index or a reference in the sequence (or checking for a transcription/interpretation).

Would you like further explanation or additional steps on how to interpret this sequence or calculation? 

Here are some questions to further explore the concept:
1. How do arithmetic means apply to arithmetic sequences in general?
2. What is the difference between arithmetic and geometric means?
3. How does this arithmetic mean relate to the rest of the sequence?
4. Can the sequence's terms be negative, and how does that affect the mean?
5. What happens if you take a larger or smaller segment of the sequence?

**Tip:** Always simplify fractions carefully in arithmetic sequences to avoid calculation errors.

واسطه حسابی بین جملات بیست و یکم و نود و سوم از دنباله حسابی، 1، -1، ... کدام است؟

This problem asks students to find the arithmetic mean between terms from a sequence. Key concepts include sequences and the arithmetic mean formula. It’s suitable for Grades 9-12 students who are learning about sequences and means.

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