Answer to Question #121413 in Algebra for queen

Question

A special event organizer wants to get the word out to 100 000 teens in the area about a concert coming up in three months. By posting the event on the musician’s website, 100 teens in the area will find out about the event in the first week. Research suggests that each of those teens will tell four new teens about the event, and those friends each will tell four more teens about it. How many times does this pattern need to continue until 100 000 teens know about the event?

Answer 1

Let S_n be a sequence that denotes the number of teens reached. Then S_n is a geometric sequence with ratio r=4 and a=100.Thus,S_n is of the form:
       S_n ={100,400,1600,...}
The general formulae for a geometric sequence is:
       S_n =ar^(n-1)
Thus to reach 100000 teens:
       100000=100*4^(n-1)
         1000=4*(n-1)
  Log_10 1000=(n-1)Log_10 4
  3/Log_10 4 = n-1
          n-1=4.98
Thus the pattern must repeat itself 5 times to reach 100000 teens