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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
 #1
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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 $\boxed{\sum_{k=0}^n 3^n}.$

 Feb 28, 2021

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