You ask your friends to help you spread the word about your upcoming picnic that you are
hosting at your house. You give an invitation to each of your three best friends. They in turn each give
invitations to three more friends. How many friends will have received an invitation after the fourth level of
friends has been invited?
level 1: 3
level 2: 3*3 = 3^2
level 3: 3^2*3 = 3^3
This is just a geometric progression, and you want the sum of the first four terms.
To find the number of friends who will have received an invitation after the fourth level of friends has been invited, we can use a simple calculation.
Level 1: You invite 3 best friends => 3 friends
Level 2: Each of your 3 best friends invites 3 more friends => 3 friends * 3 = 9 friends
Level 3: Each of the 9 friends from level 2 invites 3 more friends => 9 friends * 3 = 27 friends
Level 4: Each of the 27 friends from level 3 invites 3 more friends => 27 friends * 3 = 81 friends
Therefore, after the fourth level of friends has been invited, a total of 3 + 9 + 27 + 81 = 120 friends will have received an invitation.
To get the answer, you need to find the total number of friends invited at each level and sum them up. Let's break it down:
Level 1:
You invite your three best friends.
Level 2:
Each of your three best friends invites three more friends. So, each of them invites 3 friends. Since you have 3 best friends, the total number of friends invited in level 2 is 3 * 3 = 9.
Level 3:
Each of the 9 friends from level 2 invites three more friends. So, each of them invites 3 friends. Since there are 9 friends, the total number of friends invited in level 3 is 9 * 3 = 27.
Level 4:
Each of the 27 friends from level 3 invites three more friends. So, each of them invites 3 friends. Since there are 27 friends, the total number of friends invited in level 4 is 27 * 3 = 81.
To find the total number of friends invited after the fourth level, you sum up the numbers from each level:
Level 1: 3 friends invited
Level 2: 9 friends invited
Level 3: 27 friends invited
Level 4: 81 friends invited
Total number of friends invited after the fourth level = 3 + 9 + 27 + 81 = 120.
Therefore, after the fourth level of friends has been invited, a total of 120 friends will have received an invitation.