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He sends an e-mail to 3 of his friends (call this round 1).

Each of these friends forwards the e-mail to 3 other people (round 2).

Those 9 people each forward the e-mail to 3 other people (round 3), and so on.

Assume that nobody receives the e-mail more than once.

Determine an expression for the total number of people, including Elijah, who are aware of the good news by the nth round.

Feb 28, 2021

Since each person forwards to email to 3 other people, the expression for the number of additional people who received it after $n$ rounds (including Elijah) is $3^0 + 3^1 + 3^2 + 3^3 + \dots + 3^n.$ We can rewrite this as $\boxed{\sum_{k=0}^n 3^n}.$