A large company has a phone tree to contact its employees in case of an emergency factory shutdown. Each of the five senior managers calls three eployees, who calls three other employees and so on. If the tree consists of seven levels, how many employees does the company have?
The phone tree has seven levels, with each level having a total of 3^n employees, where n represents the level number (starting from 0).
In this case, since there are seven levels, we can calculate the number of employees in each level:
Level 0: 3^0 = 1 employee
Level 1: 3^1 = 3 employees
Level 2: 3^2 = 9 employees
Level 3: 3^3 = 27 employees
Level 4: 3^4 = 81 employees
Level 5: 3^5 = 243 employees
Level 6: 3^6 = 729 employees
Level 7: 3^7 = 2187 employees
To find the total number of employees in the company, we need to sum up the number of employees in each level:
Total = 1 + 3 + 9 + 27 + 81 + 243 + 729 + 2187
Total = 4278
Therefore, the company has a total of 4278 employees.
To solve this problem, we can use the concept of geometric progression. In this case, the number of employees at each level follows a geometric progression where the common ratio is 3.
Let's start by calculating the number of employees at the highest level, which is the number of senior managers. We are told that there are 5 senior managers.
At the next level, each senior manager contacts 3 employees, so there are 5 * 3 = 15 employees.
At the third level, each of the 15 employees contacts 3 other employees. So the total number of employees at this level is 15 * 3 = 45.
Similarly, at each subsequent level, the number of employees triples. Therefore, the number of employees at levels 4, 5, 6, and 7 can be calculated as follows:
Level 4: 45 * 3 = 135 employees
Level 5: 135 * 3 = 405 employees
Level 6: 405 * 3 = 1215 employees
Level 7: 1215 * 3 = 3645 employees
Now, to find the total number of employees in the company, we need to add up the number of employees at each level:
Total employees = 5 (senior managers) + 15 (level 2) + 45 (level 3) + 135 (level 4) + 405 (level 5) + 1215 (level 6) + 3645 (level 7)
Total employees = 5 + 15 + 45 + 135 + 405 + 1215 + 3645 = 5475
Therefore, the company has a total of 5475 employees.