Question

The president of a secret society of 65 members calls an emergency meeting in 2 days. The president calls the secretary and tells him the meeting time and place. The secretary calls 4 members and asks each of them to call 4 more members, and so on until each of the 65 members has been notified. What is the greatest number of members who do not need to make a phone call?

a) 1
b) 2
c) 3
d) 4

Answer 1

The greatest number of members who do not need to make a phone call is 20.

Step-by-step explanation:

First, let's determine the total number of members who need to make a phone call to notify others. The president calls 4 members, and each of those 4 members calls 4 more members. This process continues until each of the 65 members has been notified. We can think of this as a sequence of powers of 4: 4^1 + 4^2 + 4^3 + ... + 4^n, where n is the number of times the process is repeated. We need to find the value of n that results in a sum greater than or equal to 65.

Starting with n = 0, the sum is 4^0 = 1. Since this is less than 65, we try n = 1. The sum is 4^0 + 4^1 = 1 + 4 = 5, still less than 65. We continue this process until we find the smallest value of n that gives a sum greater than or equal to 65.

With n = 3, the sum is 4^0 + 4^1 + 4^2 + 4^3 = 1 + 4 + 16 + 64 = 85, which is greater than 65. However, we only need 65 members to be notified. So, the greatest number of members who do not need to make a phone call is 85 - 65 = 20.