1. "A bacteria population doubles in number every 3 hours. If there are 40 individuals now, how many in 24 hours?" SO I used the geometric series sum formula, with n=8(24/3=8), t1=40, and r=2, but I got 10200 as the answer, instead of what the answer key solves as 10240. What did I do wrong?
2. A runner starts jogging 4 kilometres. He increases his distance by 10% each time he runs. a) the distance he runs on the 8th day b) the total distance he has run after 8 days
I have no idea how to do this.
you don't need to sum the series. It doubles 8 times in 24 hours, so the population is 40*2^8.
You'd only need the sum if a new population, double the old one, was added every 3 hours.
The runner problem is just the same. Every day you multiply the distance by 1.1
So, on the nth day he runs 4 * 1.1^(n-1)
The sum is the series where
a = 40
r = 1.1
Sn = 40(1.1^n - 1)/(1.1 - 1)
oh I see, thank you Steve! I did everything except I didn't realize the 1.1 part! I did 0.1 instead haha
1. For the first question, you made a small error in your calculation. Let's go through the solution again using the correct formula for a geometric series:
The formula for the sum of a geometric series is: Sn = a * (1 - r^n) / (1 - r)
Given:
a = 40 (initial population size)
r = 2 (doubling factor)
n = 8 (number of intervals of 3 hours in 24 hours)
Now, let's plug the values into the formula:
Sn = 40 * (1 - 2^8) / (1 - 2)
= 40 * (1 - 256) / (1 - 2)
= 40 * (-255) / (-1)
= -40 * 255
= -10200
It seems that you have calculated the answer correctly, but you made a sign error. The answer should be -10200, not 10200. Therefore, you got the same answer as the provided solution (10240), but with the opposite sign.
2. For the second question about the runner's distance, let's calculate it step by step:
a) The distance the runner runs on the 8th day:
Start with 4 kilometers, and increase it by 10% each day for 8 days.
Distance on the 1st day = 4 km
Distance on the 2nd day = 4 km + (10% of 4 km) = 4 km + 0.4 km = 4.4 km
Distance on the 3rd day = 4.4 km + (10% of 4.4 km) = 4.4 km + 0.44 km = 4.84 km
Following this pattern, we can calculate the distance for the 8th day using the same formula repeatedly. It's important to note that we are using the previous day's distance as a base each time.
b) The total distance the runner has run after 8 days:
To calculate the total distance, you need to sum up the distances run on each day from day 1 to day 8.
Total Distance = Distance on Day 1 + Distance on Day 2 + Distance on Day 3 + ... + Distance on Day 8
I hope this explanation helps you solve the problem more effectively!