# 1. A town has a population of 100,000 now and the population is increasing 10% every year. What will be the population in 5 years?

2. How many poles will be in a pile of telephone poles if there are 30 in the bottom layer, 29 in the second, and so on, until there is one in the top layer?

3. 12. How far up and down will a ball travel before stopping if it is dropped from a height f 12 m, and each rebound is 1/3 of the previous distance.

## 1. P = Po + Po*r*t

P = 100,000 + 100,000*0.1*5 = 150,000

## 1. To find the population in 5 years, we can use the formula for compound interest: Population = Initial Population * (1 + Growth Rate)^Number of Years. In this case, the initial population is 100,000, the growth rate is 10% (or 0.1), and the number of years is 5. Plugging in these values into the formula, we get: Population = 100,000 * (1 + 0.1)^5. Evaluating this expression, we find that the population in 5 years will be approximately 161,051.

2. The number of poles in each layer form a series that decreases by 1 each time. This is an arithmetic sequence with the first term being 30 and a common difference of -1. To find the total number of poles, we can use the formula for the sum of an arithmetic series: Sum = (n/2) * (2a + (n-1)d), where n is the number of terms, a is the first term, and d is the common difference. In this case, the first term a is 30 and the common difference d is -1. Since we start with 30 poles in the bottom layer and end with 1 pole in the top layer, the number of terms n is equal to 30. Plugging in these values, we get: Sum = (30/2) * (2*30 + (30-1)(-1)). Evaluating this expression, we find that there will be a total of 465 poles.

3. Since each rebound is 1/3 of the previous distance, we have a geometric sequence where each term is 1/3 of the previous term. The distance traveled by the ball can be calculated by summing the terms of this geometric sequence. The formula for the sum of a geometric series is: Sum = a * (1 - r^n) / (1 - r), where a is the first term, r is the common ratio, and n is the number of terms. In this case, the first term a is 12 meters, the common ratio r is 1/3, and the number of terms n is not given. To find the number of rebounds (or terms) that the ball makes before stopping, we need to determine how many times the ball rebounds until the distance becomes negligible. Since each rebound is 1/3 of the previous distance, the ball will stop bouncing when the distance traveled becomes very small. Let's assume that once the distance traveled becomes less than 0.01 meters, we consider it negligible. Starting with a distance of 12 meters, we can keep multiplying the distance by 1/3 until it becomes less than 0.01 meters. Evaluating this iteration, we find that the ball will make approximately 79 rebounds before stopping. Now we can plug these values into the formula to find the total distance traveled: Sum = 12 * (1 - (1/3)^79) / (1 - (1/3)). Evaluating this expression, we find that the ball will travel approximately 17.9999 meters up and down before stopping.