Exponential Growth of emails

(ChatGPT)

Assuming perfect forwarding (each person emails 50 new people, no duplicates, everyone has a reachable email address, and sending happens in daily waves), the reach grows by a factor of 50 per day.

Using a U.S. population of about 343,121,356 (U.S. Census Bureau Population Clock, Jan 19, 2026)  :

Daily reach (new people reached each day)

• Day 1: 50
• Day 2: 50² = 2,500
• Day 3: 50³ = 125,000
• Day 4: 50⁴ = 6,250,000
• Day 5: 50⁵ = 312,500,000
• Day 6: 50⁶ = 15,625,000,000 (more than enough)

Total reached by end of each day (cumulative)

Total after day d is:

50 + 50^2 + \cdots + 50^d \;=\; 50\cdot\frac{50^d – 1}{49}

• End of Day 5: 50\cdot\frac{50^5-1}{49} \approx 318,877,550 (still short of ~343M)
• End of Day 6: 50\cdot\frac{50^6-1}{49} \approx 15,943,877,550 (exceeds the nation)

Result: Under those idealized assumptions, it takes 6 days (6 sending waves) for “the whole nation” to get an email.

In reality, duplicates, non-deliverable addresses, spam filtering, and uneven participation slow this dramatically—but the ideal math answer is 6 days.