Question 52438: 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:
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b) Using the formula for the nth term of a geometric sequence, what is the 24th term?
Answer:
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c) Using the formula for the sum of a geometric series, what is the sum of the first 10 terms?
Answer:
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Answer by venugopalramana(3286)   (Show Source):
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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
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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
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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
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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
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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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