# in a text sharing game, one teen sends text message to three teens int urn are each expected to send to 3 others,. By the 10th level , how many teens have been sent message assuming all are...

in a text sharing game, one teen sends text message to three teens int urn are each expected to send to 3 others,. By the 10th level , how many teens have been sent message assuming all are different? Draw a tree diagram for first three levels. REmember first teen was never sent a message.

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The partial tree diagram is in the attachment. As you can tell, the tree diagram can get a bit large with 59049 teens involved.

It may be easier to determine the number if we make an XY table. I will be assuming that the 3 teens initially being sent the text is the first level. X will be the level, Y will be the number of teens have the text:

X - 1, 2, 3, 4, 5, 6, 7, 8, 9, 10

Y - 3, 9, 27, 81, 243, 729, 2187, 6561, 19683, 59049

So, at the first level, it's been spread to no teens, 3 teens at the first level, etc.

From there, you can tell that it is an geometric sequence that gets multiplied by 3 each time. So:

y = 3^x

You can get the formula by comparing using the general form for an geometric sequence:

y = y(0)*b^x

where y(0) in the initial amount and b is the multiplier. Here, y(0) = 1 since we started with 1 teen having the message. So:

y = 1*3^x

y = 3^x

I analyzed the situation and manually computed until the 10th level.

1 * 3 = 3 Level 1

3 * 3 = 9 Level 2

9 * 3 = 27 Level 3

27 * 3 = 81 Level 4

81 * 3 = 243 Level 5

243 * 3 = 729 Level 6

729 * 3 = 2,187 Level 7

2,187 * 3 = 6,561 Level 8

6,561 * 3 = 19,683 Level 9

19,683 * 3 = 59,049 Level 10

The total number of teenager after the 10th level would be **59, 049**