Problem

To determine the average afternoon wait time in s…

01:53

Problem 116 Hard Difficulty

Chain Letters. A woman sent two friends a letter with the following request: "Please send a copy of this letter to two of your friends." Assume that all those receiving letters responded and that everyone in the chain received just one letter. Complete the table and then determine how many letters will be circulated in the 10 th level.
$$
\begin{array}{|c|c|}
\hline \text { Level } & {\text { Number of letters circulated }} \\
\hline \text { 1st } & {2=2^{1}} \\
\hline 2 \mathrm{nd} & {=2} \\
\hline 3 \mathrm{rd} & {=2} \\
\hline 4 \mathrm{th} & {=2} \\
\hline
\end{array}
$$

Answer

1024 letters on 10 th level

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Video Transcript

in this problem. We're told that a woman sent two letters. Our two friends, a letter with the following request. Please send the coffee this letter to two of your friends. So we're gonna assume that everyone who gets a letter is going to respond and that everyone in the chain is only gonna get one letter. So first we want to fill in the table, and then we're gonna figure out how many letters will be circulated in the 10th level. So let's think about what we started with. You started with one person, and they're going to send this letter to two people. So here's the first person and here's the second person. So if this is considered to be our first level, and there's a total of two people receiving it, So if you know this in the first row, we have in the first level that to total people get it, which could be rewritten as two to the first power. So now let's think about what's gonna happen next. If both of these people send it there letter to a friend, this first person will send it to two people, and the second person will send it to two people. So there's gonna be a total of four letters. So for our total is equal to four. Well, this first person sent it to two people in the second person, sent it to two people, so we can think of it in terms of exponents as two to the second power. Well, now, for the third level, each of these people will also send this letter to a two people, so this person will send it to two. This wanted to this one. That too. And this last one that you So how many total will be 12345678 So in this third level, there's a total off eight letters being said Well, to tell what power is equal to eight. Well, that would be two to the third power. So hopefully you've kind of caught on to a pattern here. So now if all eight of these people send their letter to two people each, let's see if I have enough room here. There we go. If we count up, how many letters are sent will find its 16. So in the four flutter, 16 total letters were sent out well to toe what power is equal to 16. Well, that would be equal before. So let's see if we've noticed a pattern. Find a number total letter circulated. We simply take two and we raise it to D power of the same level. So if we are now being asked how many letters are sent in 1/10 level, we simply need to find two to the 10th power. Well, due to the tongue, power means you need to multiply two by itself 10 times, which is equal to 1024. So in the 10th level level.

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