int fun(int n) 
{
    int count = 0;
    for (int i = n; i > 0; i /= 2)  // Outer loop
    {
        for (int j = 0; j < i; j++)  // Inner loop
        {
            count += 1;
        }
    }
    return count;
}  find complxity

To determine the time complexity of the given function, let's analyze the loops step by step.

### Function Analysis:

```cpp
int fun(int n) 
{
    int count = 0;
    for (int i = n; i > 0; i /= 2)  // Outer loop
    {
        for (int j = 0; j < i; j++)  // Inner loop
        {
            count += 1;
        }
    }
    return count;
}
```

#### Outer Loop:
- The outer loop starts with `i = n` and in each iteration, `i` is divided by 2 (`i /= 2`).
- This loop runs as long as `i > 0`.

The value of `i` in successive iterations will be `n, n/2, n/4, n/8, ..., 1`. The number of iterations of this loop can be determined by how many times we can divide `n` by 2 before it becomes 1. This number is approximately `log₂(n)`.

So, the outer loop runs `O(log n)` times.

#### Inner Loop:
- For each iteration of the outer loop, the inner loop runs `i` times, where `i` is the current value of the outer loop variable.
- Therefore, in the first iteration, the inner loop runs `n` times, in the second iteration it runs `n/2` times, in the third iteration `n/4` times, and so on.

The total number of iterations of the inner loop across all iterations of the outer loop is the sum of a geometric series:

\[
\text{Total number of inner loop iterations} = n + \frac{n}{2} + \frac{n}{4} + \frac{n}{8} + \dots + 1
\]

This is a geometric series with the first term as `n` and the common ratio `1/2`. The sum of this series is given by:

\[
S = \frac{n}{1 - \frac{1}{2}} = 2n
\]

So, the total number of operations performed by the inner loop is `2n`.

### Time Complexity:
- The outer loop runs `O(log n)` times.
- The inner loop runs for a total of `O(n)` times across all iterations.

Therefore, the overall time complexity of the function is:

\[
O(2n) \approx O(n)
\]

**Final Answer:** The time complexity of the function is **O(n)**.

---

Do you want further details or have any questions?

Here are some related questions:
1. How does the inner loop's time complexity change if the increment in `j` was different?
2. What would the complexity be if the outer loop had a different stopping condition?
3. How does a geometric series influence the complexity of nested loops?
4. How would the complexity change if the outer loop had a different step decrement, like `i -= 1`?
5. What impact does using `i /= 3` instead of `i /= 2` have on the time complexity?

**Tip:** Understanding how nested loops interact, particularly when the loop variables depend on each other, is crucial for analyzing time complexity efficiently.

Analyze the time complexity of a nested loop function using concepts like algorithm analysis and geometric series. This example breaks down the process to determine that the time complexity of the function is O(n). Ideal for undergraduate computer science students.

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