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Question 1003729: The sum of the infinite geometric sequence 2 - 1/5 + 1/25 - 1/125 + .... is ______.
A. 14/5
B. 6/5
C. 6/13
D. 11/6
E. 9/4
Answer by jim_thompson5910(35256)   (Show Source):
You can put this solution on YOUR website! We have the first term, which is a = 2. We need the common ratio r.
Let's divide the second term over the first term
r = (second term)/(first term)
r = (-1/5)/(2)
r = (-1/5)*(1/2)
r = -1/10
Let's divide the third term over the second term
r = (third term)/(second term)
r = (1/25)/(-1/5)
r = (1/25)*(-5/1)
r = -5/25
r = -1/5
Notice how r changes value. Since r is NOT the same each time, this means that this sequence is NOT a geometric sequence. So there has to be a typo somewhere. Please double check the problem. Thank you.
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Edit: I realize that the sequence " - 1/5 + 1/25 - 1/125 + ..." is geometric (ignore the 2 up front). The first term of this sequence is a = -1/5 . The common ratio is...
r = (second term)/(first term)
r = (1/25)/(-1/5)
r = (1/25)*(-5/1)
r = -5/25
r = -1/5
Since |r| < 1, the infinite series converges to a fixed number, which is
S = a/(1-r)
S = (-1/5)/(1-(-1/5))
S = (-1/5)/(1+1/5)
S = (-1/5)/(6/5)
S = (-1/5)*(5/6)
S = -1/6
So 2 - 1/5 + 1/25 - 1/125 + .... = 2 + (- 1/5 + 1/25 - 1/125 + ....) = 2 + (-1/6) = 11/6
Final answer: 11/6
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