A boy had e-mailed a message for four friends, then each friend e-mailed the same message to another four friends each of whom did the same, where each friend received the message once only. Determine the series using the summation notation, then find the total number of people who received the message by the sixth wave.
Initially, the email was received by one person . Then on each iteration, every person who has received the email sends it to four people. We express this mathematically as follows: . Consequently,
Hence, we can express the iteration as . Therefore, assuming the series never ends, we can express it in summation notation as follows:
To calculate the number of people who will have received the message by the sixth wave, we evaluate the series up to the sixth term: