Answer: We will never reach a sum of 2
We get closer and closer to 2, but never actually reach this exact value.
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Step-by-step explanation:
The sequence
1, 1/2, 1/4, 1/8, 1/16, 1/32, 1/64, ...
is geometric with these properties
- a = 1 = first term
- r = 1/2 = common ratio
We multiply each term by 1/2 to get the next one.
Examples:
- (1/8)*(1/2) = 1/16
- (1/32)*(1/2) = 1/64
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Notice how -1 < r < 1 is true, i.e. -1 < 1/2 < 1 is true.
Because of this fact, we can determine the sum of infinitely many terms.
That infinite sum is
S = a/(1-r)
This is our upper bound of what we can reach for S.
Calculating it gives:
S = a/(1-r)
S = 1/(1-0.5)
S = 2
Therefore, the sum of the infinitely many terms of this geometric sequence is 2.
1 + 1/2 + 1/4 + 1/8 + 1/16 + 1/32 + 1/64 + ... = 2
We never quite reach 2 exactly due to the fact we cannot reach infinity on the number line. Infinity is not a number, but rather a concept.
Therefore, we never reach a sum of 2. We simply get closer and closer.
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Using computer software, I was able to generate this:
- 1 + 1/2 = 1.5
- 1 + 1/2 + 1/4 = 1.75
- 1 + 1/2 + 1/4 + 1/8 = 1.875
- 1 + 1/2 + 1/4 + 1/8 + 1/16 = 1.9375
- 1 + 1/2 + 1/4 + 1/8 + 1/16 + 1/32 = 1.96875
- 1 + 1/2 + 1/4 + 1/8 + 1/16 + 1/32 + 1/64 = 1.984375
The decimal values are exact without any rounding done to them. As you can see, the results of 1.5, 1.75, 1.875, etc are slowly approaching 2 but never get to that exact value itself. Visually you can think of an asymptote.