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.