A chain letter works as follows: One person sends a copy of the letter to five friends, each of whom sends a copy to five friends, each of whom sends a copy to five friends, and so forth. How many people will have received copies of the let - ter after the twentieth repetition of this process, assuming no person receives more than one copy?

The number of people that received copies of the letter at the twentieth stage is 9.537 * 10¹³.

Step-by-step explanation:

Using the discrete model,

a_k = r a_ (k-1) for all integers k ≥ 1 and a₀ = a

then,

aₙ = a rⁿ for all integers n ≥ 0

Let a_k be equal to the number of people who receive a copy of the chain letter at a stage k.

Initially, one person has the chain letter (which the person will send to five other people at stage 1). Thus,

a = a₀ = 1

The people who received he chain letter at stage (k - 1), will send a letter to five people at stage k and thus per person at stage (k - 1), five people will receive the letter. Therefore,

a_k = 5 a_ (k - 1)

Thus,

aₙ = a rⁿ = 1 · 5ⁿ = 5ⁿ

The number of people that received copies of the letter at the twentieth stage is

a₂₀ = (5) ²⁰ = 9.537 * 10¹³ copies