SOLUTION: The government of a developing country is implementing a new reading program in which each individual who can read spends the year teaching two new people how to read. At the end

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Question 471020: The government of a developing country is implementing a new reading program in which each individual who can read spends the year teaching two new people how to read. At the end of the year the individual can stop but each of the two new readers have to teach two more people to read. If 1000 people can read at the start of the program, how many people will be able to read at the end of 10 years?
I started making a table where I added the original and the new readers to get the total readers for the first year. For each subsequent year on the table I take the new readers and add them to their new readers and then add the total of readers from the previous years to get the total of readers for the new years. When I finish my table for 10 years I came up with 2047000. Don't know if that is right. Help????
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
Here's the basic table you should have:
Year x: Beginning --> End
--------------------------
year 1: 1000 --> 2000
year 2: 2000 --> 4000
year 3: 4000 --> 8000
year 4: 8000 --> 16000
year 5: 16000 --> 32000
year 6: 32000 --> 64000
year 7: 64000 --> 128000
year 8: 128000 --> 256000
year 9: 256000 --> 512000
year 10: 512000 --> 1024000
Sum = 1000+2000+4000+8000+16000+32000+64000+128000+256000+512000+1024000 = 2047000


So you are correct. Good job.


A shortcut is to realize that these values form a geometric sequence that fit the form a%5Bn%5D=a%2Ar%5E%28n-1%29 where n starts at n=1. In your case, a=1000 and r=2. So the formula is a%5Bn%5D=1000%2A2%5E%28n-1%29


Now you can use the formula S=a%28%281-r%5En%29%2F%281-r%29%29, where S is the partial sum from n=1 up to the nth term, which in this case is n=11 (note: at the END of year 10 is the start of year 11, so we sum up to 11 to include the last readers taught in year 10).


So this means that the partial sum is


So we get the same answer.