Chapter 9.1, Problem 53E

Solution

To state: How you know with certainty that you will receive a second copy of the letter later in the year if in the first week of a the year a letter is send to five addressees and then these five person sends a copy of this letter o five others during the second week of the year and it continues further.

The resultant answer is after one year.

Given information:

In the first week of the year a letter is send to five addressees and then these five person sends a copy of this letter o five others during the second week of the year and it continues further.

Explanation:

In the first week a letter send to 5 persons.

In the second week these five persons send the letter to other five then it will become: $5^2$

In the third week these $5^2$ persons send the letter to other five then it will become: $5^3$

As it is known that a year contains 52 weeks, the in 52 weeks the letter will reach $5^{52}$ persons.

  $5^{52}\approx 2.22\times {10}^{36}$

The population of the world is about $6\times {10}^9$ . It can be seen that $2.22\times {10}^{36}$ is greater that the world’s population.

From the above answer it is clear that if someone has received a copy of the letter before, then he will receive a copy again after one year.