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The problem provided in the image appears to involve a sequence \( a_n \) defined as follows:

- For **odd** \( n \): 
  \[
  a_n = \frac{n+1}{2}
  \]
- For **even** \( n \):
  \[
  a_n = \frac{n}{2}
  \]

The question asks: **"How many integers appear in the sequence twice?"**

### Let's break down the pattern:

1. **Odd \( n \):**
   - For odd \( n \), the formula is \( a_n = \frac{n+1}{2} \). For example:
     - \( a_1 = \frac{1+1}{2} = 1 \)
     - \( a_3 = \frac{3+1}{2} = 2 \)
     - \( a_5 = \frac{5+1}{2} = 3 \), and so on.

2. **Even \( n \):**
   - For even \( n \), the formula is \( a_n = \frac{n}{2} \). For example:
     - \( a_2 = \frac{2}{2} = 1 \)
     - \( a_4 = \frac{4}{2} = 2 \)
     - \( a_6 = \frac{6}{2} = 3 \), and so on.

### Observing Repeated Values:
- From the formulas, you can observe that:
  - For odd \( n \), \( a_n = \frac{n+1}{2} \)
  - For even \( n \), \( a_n = \frac{n}{2} \)
  
  These two rules will often result in the same value for two different \( n \) values (one even, one odd). For example:
  - \( a_1 = 1 \) and \( a_2 = 1 \) (both yield 1)
  - \( a_3 = 2 \) and \( a_4 = 2 \) (both yield 2)
  - \( a_5 = 3 \) and \( a_6 = 3 \), and so on.

### To find how many numbers appear twice:
- The numbers that appear twice are exactly those numbers for which both an odd and even \( n \) give the same value.
- This happens for every pair of consecutive integers. The sequence runs from \( a_1 \) to \( a_{150} \). The odd numbers and even numbers will match in this paired way up to \( a_{150} \), leading to repeated values.

### Conclusion:
- The integers from 1 to 75 appear twice because the sequence alternates between odd and even values that pair up in this way.
  
Thus, **75 integers** appear twice.

Given the answer choices (11, 10, 9, 8), it seems like there's a mismatch or misprint in the provided answer options, but logically, **75 integers** should appear twice in this sequence.

Would you like more details or have any further questions? Here are some related questions:

1. How would you determine the largest integer in a similar sequence defined differently?
2. What is the general pattern for sequences involving odd/even differences?
3. How can you model sequences with different recurrence relations?
4. How would the sequence change if the formulas for odd/even terms were switched?
5. How would you approach finding the sum of the first \( n \) terms in this sequence?

**Tip:** Sequences like this often alternate between formulas depending on parity (odd/even) of \( n \). Identifying patterns early can help solve them more efficiently.

The nth term of the sequence shown is defined for each integer n from 1 to 150 as follows: If n is odd, then a_n = (n+1)/2, and if n is even, a_n = n/2. How many integers appear in the sequence twice?

This problem explores how a sequence defined by different rules for odd and even numbers creates repeated integers. Learn how to identify patterns in arithmetic sequences for integers from 1 to 150, using the formulas a_n = (n+1)/2 and a_n = n/2 for odd and even numbers, respectively. Suitable for high school students.

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