Problem:
The rock group Loco Moco was scheduled for a concert at the Blaisdell Concert Hall. Because it was the concert highlight of the year, there was excitement in the air and people were eager to get tickets, which could only be purchased in person. Hours before tickets were to go on sale, people were lined up to buy tickets. In fact, the first person came 12 hours before the ticket booth was open. A new group of ticket buyers joined the line every 30 minutes.
Mission:
If each new group has four persons more than the previous group, how many people were in line after the 20th group joined, assuming the first person is the first group?
If each new group had two persons more, how do you think the number in line after the 20th group joined compares to the number of people line when four people joined as described in problem 1?
I need a pattern
To find a pattern in the number of people in line after each group joins, let's break down the problem and calculate the number of people in line at each step.
Problem 1:
Each new group has four persons more than the previous group.
Step 1: The first person came 12 hours before the ticket booth was open, so there was only 1 person in the line.
Step 2: The second group joins the line, adding 4 persons to the previous group, so there are 1 + 4 = 5 people in line.
Step 3: The third group joins, adding another 4 persons to the previous group, so there are 5 + 4 = 9 people in line.
Step 4: The fourth group joins, adding 4 more persons, making it 9 + 4 = 13 people in line.
Step 5: The fifth group joins, adding 4 more persons, making it 13 + 4 = 17 people in line.
And so on...
To find the number of people in line after the 20th group joins, we need to continue this pattern up to the 20th step.
Step 6: 17 + 4 = 21 people in line.
Step 7: 21 + 4 = 25 people in line.
Step 8: 25 + 4 = 29 people in line.
...
Step 20: 13 + (4 * 19) = 13 + 76 = 89 people in line.
So, after the 20th group joins, there are 89 people in line.
Problem 2:
Each new group has two persons more than the previous group.
Using the same logic, we can calculate the number of people in line after the 20th group joins.
Step 1: 1 person in line.
Step 2: 1 + 2 = 3 people in line.
Step 3: 3 + 2 = 5 people in line.
Step 4: 5 + 2 = 7 people in line.
...
Step 20: 1 + (2 * 19) = 1 + 38 = 39 people in line.
Comparing Problem 1 and Problem 2, we see that the number of people in line after the 20th group joins is higher in Problem 1 (89 people) than in Problem 2 (39 people) because each new group adds more people in Problem 1.
Pattern:
In this problem, each new group joining the line follows a pattern where the size of the group increases by a certain number of persons in each subsequent group.
To find the pattern, we need to understand how the size of each new group is determined.
1. The first group consists of one person.
2. Each new group after the first group has four persons more than the previous group.
Now let's calculate the number of people in line after the 20th group joined for both scenarios.
Problem 1:
If each new group has four persons more than the previous group:
To find the number of people in the line after the 20th group joined, we need to calculate the size of each group and sum them up.
Group 1: 1 person
Group 2: 1 + 4 = 5 persons
Group 3: 5 + 4 = 9 persons
Group 4: 9 + 4 = 13 persons
...
We can see that each group's size increases by 4 each time. So, the size of the 20th group would be:
Group 20: 1 + (20 - 1) * 4 = 1 + 19 * 4 = 1 + 76 = 77 persons
Therefore, after the 20th group joined, there would be 77 people in line.
Problem 2:
If each new group had two persons more than the previous group:
Similar to problem 1, let's calculate the number of people in line after the 20th group joined.
Group 1: 1 person
Group 2: 1 + 2 = 3 persons
Group 3: 3 + 2 = 5 persons
Group 4: 5 + 2 = 7 persons
...
In this case, each group's size increases by 2 each time. So, the size of the 20th group would be:
Group 20: 1 + (20 - 1) * 2 = 1 + 19 * 2 = 1 + 38 = 39 persons
Therefore, after the 20th group joined, there would be 39 people in line.
Comparison:
From the calculations above, we can see that in problem 2, where each new group has two persons more than the previous group, there would be fewer people in line after the 20th group joined (39 persons) compared to problem 1, where each new group has four persons more than the previous group (77 persons).