Answer
\[
\boxed{5 + 5^2 + 5^3 + \cdots + 5^{20}
\;=\;
5\,\frac{5^{20} - 1}{4}.
}
\]
Work Step by Step
We have a chain‐letter process where:
- One person (the origin) sends the letter to 5 new people.
- **Each** of those 5 recipients sends the letter to 5 **new** people.
- That pattern continues, with each new recipient sending it to 5 new people, etc.
We want the total number of **distinct** recipients of the letter **after the twentieth repetition** (wave).
---
## How to Count the Recipients
- **Wave 1**: The original sender mails out 5 copies.
So \(5\) new people receive the letter in the first wave.
- **Wave 2**: Those 5 each send to 5 new people, so \(5^2 = 25\) more receive it.
- **Wave 3**: Those \(25\) each send to 5 new people, so \(5^3 = 125\) more receive it.
- \(\dots\)
- **Wave \(k\)**: \(5^k\) new people receive it.
After \(n\) waves (repetitions), the **total** who have received the letter is
\[
5 + 5^2 + 5^3 + \cdots + 5^n.
\]
In particular, after the **20th** repetition:
\[
\text{Total recipients}
\;=\; \sum_{k=1}^{20} 5^k.
\]
---
## Closed‐Form Formula
The sum of a geometric series
\(\sum_{k=1}^{n} r^k = r \,\frac{r^n - 1}{r - 1}\).
Here, \(r = 5\). Thus,
\[
\sum_{k=1}^{20} 5^k
\;=\; 5 \,\frac{5^{20} - 1}{5 - 1}
\;=\; 5 \,\frac{5^{20} - 1}{4}.
\]
That is the exact number of people who have received the letter after 20 waves (not counting the original sender, since they are not a “recipient”).
If a simplified answer is acceptable, we can simply write:
\[
\boxed{5 + 5^2 + 5^3 + \cdots + 5^{20}
\;=\;
5\,\frac{5^{20} - 1}{4}.
}
\]
Either form is correct, but the factored version is often how geometric sums are presented.
