<PRE><B>Question 52438</B>
SEE THE FOLLOWING EXAMPLE AND TRY.IF STILL IN DIFFICULTY PLEASE COME BACK
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    Use the geometric sequence of numbers 1, 2, 4, 8,…to find the following:
a)   What is r, the ratio between 2 consecutive terms? 
Answer:
 
r = 2

Show work in this space.          


b)   Using the formula for the nth term of a geometric sequence, what is the 24th term?
Answer:            


      a24 = 8388608
 
Show work in this space.
 
an = 2n-1
a24 = 224-1
a24 = 223
a24 = 8388608  


c)   Using the formula for the sum of a geometric series, what is the sum of the first 10 terms? 
Answer: 
 
Sum a10 = 1023

Show work in this space           

Sum an = (2n - 1) / ( 2 - 1 )
Sum a10 = ( 210 - 1 ) / ( 2 - 1 )
      Sum a10 = ( 210 - 1 ) / ( 1 )
Sum a10 = ( 1024 - 1 ) / ( 1 )
Sum a10 = ( 1023 ) / ( 1 )
Sum a10 = 1023
 



3)         Use the geometric sequence of numbers 1, 1/2, 1/4, 1/8,…to find the following:
a)   What is r, the ratio between 2 consecutive terms? 
Answer:
 
r = 1/2
                  
Show work in this space.          





b)   Using the formula for the sum of the first n terms of a geometric series, what is the sum of the first 10 terms? Please round your answer to 4 decimals.
Answer: 
 
a9 = 1.9980

Show work in this space.
 
Sum an = (.5n+1 - 1) / ( .5 - 1 )
 
            Starts at 0 so a9 is 10 terms.
Sum a9 = (.59+1 - 1) / ( .5 - 1 )
Sum a9 = (.00195 - 1) / ( -.5 )
Sum a9 = (-.99002) / ( -.5 )
a9 = 1.9980
 
c)   Using the formula for the sum of the first n terms of a geometric series, what is the sum of the first 12 terms? Please round your answer to 4 decimals.
Answer:            


a10 = 1.9995
 
Show work in this space.          

Sum an = (.5n+1 - 1) / ( .5 - 1 )
 
            Starts at 0 so a11 is 12 terms.
Sum a11 = (.511+1 - 1) / ( .5 - 1 )
Sum a11 = (.000244 - 1) / ( -.5 )
Sum a11 = (-.99976) / ( -.5 )
a9 = 1.9980
a10 = 1.9995


 
 
d)   What observation can make about these sums? In particular, what number does it appear that the sum will always be smaller than?
Answer:            

The Sum is always smaller than 2.
 



N a Sum 
0 1 1 
1 0.5 1.5 
2 0.25 1.75 
3 0.125 1.875 
4 0.0625 1.9375 
5 0.03125 1.96875 
6 0.015625 1.984375 
7 0.007813 1.992188 
8 0.003906 1.996094 
9 0.001953 1.998047 
10 0.000977 1.999023 
11 0.000488 1.999512 
12 0.000244 1.999756 

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