To find the number of terms in the arithmetic sequence $a = \{ \frac{1}{2}, 2, \frac{7}{2}, \ldots, 11 \}$, we proceed as follows:

Step 1: Identify the common difference ($d$)

The first term ($a_1$) is $\frac{1}{2}$.

The second term ($a_2$) is $2$.

The common difference ($d$) is calculated as: $d = a_2 - a_1 = 2 - \frac{1}{2} = \frac{4}{2} - \frac{1}{2} = \frac{3}{2}.$

So, $d = \frac{3}{2}$.

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Step 2: Use the formula for the $n$-th term of an arithmetic sequence

The formula for the $n$-th term of an arithmetic sequence is: $a_n = a_1 + (n-1)d$

Substitute $a_1 = \frac{1}{2}$, $d = \frac{3}{2}$, and $a_n = 11$ (the last term): $11 = \frac{1}{2} + (n-1) \cdot \frac{3}{2}.$

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Step 3: Solve for $n$

1. Subtract $\frac{1}{2}$ from both sides: $11 - \frac{1}{2} = (n-1) \cdot \frac{3}{2}.$

Simplify: $\frac{22}{2} - \frac{1}{2} = (n-1) \cdot \frac{3}{2},$ $\frac{21}{2} = (n-1) \cdot \frac{3}{2}.$

2. Multiply through by 2 to eliminate the denominator: $21 = 3(n-1).$

3. Divide through by 3: $n-1 = 7.$

4. Add 1 to both sides: $n = 8.$

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Final Answer:

The number of terms in the sequence is 8.

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Would you like a detailed explanation of any step or additional examples?

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Related Questions:

1. How do you calculate the sum of this arithmetic sequence?
2. What are the formulas for the common difference and the $n$-th term?
3. Can you derive the sum of the first $n$ terms of any arithmetic sequence?
4. What is the difference between arithmetic and geometric sequences?
5. How does one verify that a sequence is arithmetic?

Tip:

When working with fractions in sequences, always simplify and work with the least common denominator (LCD) to avoid errors.