(a) Find the sum of the first 200 natural numbers. (b) A golfball is dropped from a height of 30 ft to the pavement. It always rebounds three fourths of the (4) distance that it drops. How far (up and down) will the ball have traveled when it hits the pavement for the 6th time?

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### Problem (a)

**Find the sum of the first 200 natural numbers.**

The sum of the first $$ n $$ natural numbers can be calculated using the formula:

$$
S = \frac{n(n + 1)}{2}
$$

For $$ n = 200 $$:

$$
S = \frac{200 \times 201}{2} = 100 \times 201 = 20,\!100
$$

**Answer:** The sum of the first 200 natural numbers is **20,100**.

---

### Problem (b)

**A golf ball is dropped from a height of 30 ft to the pavement. It always rebounds three fourths of the distance it drops. How far (up and down) will the ball have traveled when it hits the pavement for the 6th time?**

Let's break down the ball's journey step-by-step:

1. **Initial Drop:**
   - **Down:** 30 ft (hits the pavement for the 1st time).

2. **Subsequent Bounces:**
   - After each hit, the ball rebounds to $$ \frac{3}{4} $$ of the previous drop height and then falls back the same distance.
   - This sequence continues until the 6th hit.

Let's calculate each segment:

| Hit Number | Upward Distance | Downward Distance | Cumulative Distance |
|------------|------------------|--------------------|---------------------|
| 1          | —                | 30 ft              | 30 ft                |
| 2          | $$ \frac{3}{4} \times 30 = 22.5 $$ ft | 22.5 ft | $$ 30 + 22.5 + 22.5 = 75 $$ ft       |
| 3          | $$ \frac{3}{4} \times 22.5 = 16.875 $$ ft | 16.875 ft | $$ 75 + 16.875 + 16.875 = 108.75 $$ ft    |
| 4          | $$ \frac{3}{4} \times 16.875 = 12.65625 $$ ft | 12.65625 ft | $$ 108.75 + 12.65625 + 12.65625 = 134.0625 $$ ft |
| 5          | $$ \frac{3}{4} \times 12.65625 = 9.4921875 $$ ft | 9.4921875 ft | $$ 134.0625 + 9.4921875 + 9.4921875 = 153.046875 $$ ft |
| 6          | $$ \frac{3}{4} \times 9.4921875 = 7.119140625 $$ ft | 7.119140625 ft | $$ 153.046875 + 7.119140625 + 7.119140625 = 167.28515625 $$ ft |

**Total Distance Traveled:** Approximately **167.29 feet** when the ball hits the pavement for the 6th time.

**Answer:**  
(a) The sum is 20,100.  
(b) The ball travels approximately 167.29 ft up and down by the 6th hit.
(a) The sum of the first 200 natural numbers is **20,100**. (b) The golf ball travels approximately **167.29 feet** up and down by the 6th hit.

(a) Find the sum of the first 200 natural numbers. (b) A golfball is dropped from a height of 30 ft to the pavement. It always rebounds three fourths of the (4) distance that it drops. How far (up and down) will the ball have traveled when it hits the pavement for the 6th time?

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