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?
Could someone explain to me how to solve this? Which geometric formula would I have to use?
s7=5(3^7-1)/3-1
=5465
Use the formula
Sn = a(r^n-1)
--------
r-1
THE first layer 5
second layer 5x3
third layer 5x3x3
see the pattern?5*3^(n-1)
where n is the layer.
employees total is the sum of all of these. Looks like a geometric series..
If you call the five senior managers as the first level, then in the second level there would be 15 and in the third level there would be 45
so you want the 7th term in this sequence, not the sum of 7 terms (because you would be counting the same people again and again)
so a=5, r=3, and t(7) = ar^6 = 5(3)^6 = 3645
check my arithmetic.
Well, to be honest, it sounds like this company has quite a complex network of communication going on. But don't worry, I'm here to help you figure it out, with a touch of humor, of course!
Now, let's break it down like a clown riding a tiny bicycle. At each level, each employee reaches out to three others. So, if we start at level one with the five senior managers, on level two each of them reaches three employees, leading to a total of 5 x 3 = 15 employees.
Now, as the phone tree shoots off like confetti cannons, on level three, each of those 15 employees calls three more employees, resulting in 15 x 3 = 45 employees.
Following this pattern, level four will have 45 x 3 = 135 employees, level five will have 135 x 3 = 405 employees, level six will have 405 x 3 = 1215 employees, and finally, level seven will have 1215 x 3 = 3645 employees.
Now, if we add up all the employees at each level, we get a grand total of 5 (senior managers) + 15 + 45 + 135 + 405 + 1215 + 3645 = 5475 employees.
So, drumroll, please! The company has a whopping 5475 employees!
Keep in mind that this calculation assumes that each employee is only contacted once in the phone tree. If there are any repeated or excluded employees, well, then we might need to call in some juggling clowns to sort things out!
To solve this problem, you can use the geometric formula for the sum of a geometric series. In this case, the series represents the total number of employees at each level of the phone tree.
To start, you need to determine the initial number of employees at level one, which is given as five senior managers. Let's call this number "a" (a = 5).
Next, you need to find the common ratio, which is the number of employees each person calls. In this case, each person calls three other employees. Let's call this number "r" (r = 3).
Now, you can use the formula for the sum of a geometric series:
S = a * (1 - r^n) / (1 - r)
S represents the total number of employees in the phone tree, n is the number of levels, "a" is the initial number of employees, and "r" is the common ratio.
In this case, n = 7, a = 5, and r = 3.
Plugging in the values into the formula, we have:
S = 5 * (1 - 3^7) / (1 - 3)
Now, we can simplify this expression:
S = 5 * (1 - 2187) / (1 - 3)
= 5 * (-2186) / (-2)
= 5 * 1093
= 5465
Therefore, the company has a total of 5465 employees.
By using the formula for the sum of a geometric series, you can easily calculate the total number of employees in this problem.