A man tries to ascend a greased pole 101 metres high. He ascends 10 metres in first minutes and slips down 1 metre in the alternate minute. If he continues to ascends in this way. How long does he takes to reach the top ?

Answer

To solve the problem of how long it takes for the man to ascend a greased pole of 101 meters, we need to analyze his climbing pattern over time.

### Step-by-Step Solution:

1. **Understanding the Climbing Pattern**:
- In the **first minute**, the man ascends **10 meters**.
- In the **second minute**, he slips down **1 meter**.
- Therefore, in **2 minutes**, his net gain is \(10 - 1 = 9\) meters.

2. **Calculating the Effective Gain**:
- For every **2 minutes**, he effectively climbs **9 meters**.
- We can calculate how many complete cycles of 2 minutes it takes to get close to 101 meters.

3. **Finding the Number of Cycles**:
- Let’s see how many complete cycles of 2 minutes it takes to reach just below 101 meters.
- After \(n\) cycles of 2 minutes, he climbs \(9n\) meters.
- We need to find \(n\) such that \(9n < 101\).
- The maximum \(n\) satisfying this is \(n = 11\) because \(9 \times 11 = 99\) (which is less than 101) and \(9 \times 12 = 108\) (which exceeds 101).

4. **Calculating Time for Complete Cycles**:
- For \(n = 11\) cycles, the time taken is \(11 \times 2 = 22\) minutes.
- After 22 minutes, he has climbed **99 meters**.

5. **Final Climb**:
- In the **23rd minute**, he ascends **10 meters**.
- After this minute, he reaches \(99 + 10 = 109\) meters, which means he has surpassed the height of the pole.

6. **Total Time Taken**:
- He reaches the top of the pole during the **23rd minute**.
- Therefore, the total time taken to reach the top of the pole is **23 minutes**.

### Final Answer:
The man takes **23 minutes** to reach the top of the greased pole.