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03 Aug 2025, 08:47
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plpurva
Which formula are you suggesting we use, and how would it apply here? Could you please elaborate?
12 Sep 2025, 07:22
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Let me help you tackle this alternating series problem step-by-step.
Understanding the Sequence
The formula \((-1)^{(k+1)} \cdot \frac{1}{2^k}\) creates an alternating series where terms get progressively smaller. The key is recognizing how the signs alternate and the magnitudes decrease.
Step 1: Calculate the First Few Terms
For \(k = 1\): \((-1)^{(1+1)} \cdot \frac{1}{2^1} = (-1)^2 \cdot \frac{1}{2} = \frac{1}{2}\)
For \(k = 2\): \((-1) ^{(2+1)} \cdot \frac{1}{2^2} = (-1)^3 \cdot \frac{1}{4} = -\frac{1}{4}\)
For \(k = 3\): \((-1) ^{(3+1)} \cdot \frac{1}{2^3} = (-1)^4 \cdot \frac{1}{8} = \frac{1}{8}\)
For \(k = 4\): \((-1) ^{(4+1)} \cdot \frac{1}{2^4} = (-1)^5 \cdot \frac{1}{16} = -\frac{1}{16}\)
Continuing this pattern through \(k = 10\), we get:
\(\frac{1}{2}, -\frac{1}{4}, \frac{1}{8}, -\frac{1}{16}, \frac{1}{32}, -\frac{1}{64}, \frac{1}{128}, -\frac{1}{256}, \frac{1}{512}, -\frac{1}{1024}\)
Step 2: Strategic Pairing
Since we have alternating positive and negative terms, let's pair consecutive terms:
Pair 1: \(\frac{1}{2} + (-\frac{1}{4}) = \frac{2}{4} - \frac{1}{4} = \frac{1}{4}\)
Pair 2: \(\frac{1}{8} + (-\frac{1}{16}) = \frac{2}{16} - \frac{1}{16} = \frac{1}{16}\)
Pair 3: \(\frac{1}{32} + (-\frac{1}{64}) = \frac{1}{64}\)
Pair 4: \(\frac{1}{128} + (-\frac{1}{256}) = \frac{1}{256}\)
Pair 5: \(\frac{1}{512} + (-\frac{1}{1024}) = \frac{1}{1024}\)
Step 3: Calculate the Sum
\(T = \frac{1}{4} + \frac{1}{16} + \frac{1}{64} + \frac{1}{256} + \frac{1}{1024}\)
Converting to common denominator 1024:
\(T = \frac{256}{1024} + \frac{64}{1024} + \frac{16}{1024} + \frac{4}{1024} + \frac{1}{1024} = \frac{341}{1024}\)
Step 4: Compare to Answer Choices
Since \(\frac{1}{4} = \frac{256}{1024}\) and \(\frac{1}{2} = \frac{512}{1024}\)
And \(256 < 341 < 512\)
Therefore, \(\frac{1}{4} < T < \frac{1}{2}\)
Answer: D
Want to master the systematic framework for alternating series and discover alternative approaches that work even faster? Check out the complete solution on Neuron by e-GMAT, which reveals the underlying pattern that applies to all similar GMAT problems. Access detailed explanations for official questions with practice quizzes and analytics here.
Understanding the Sequence
The formula \((-1)^{(k+1)} \cdot \frac{1}{2^k}\) creates an alternating series where terms get progressively smaller. The key is recognizing how the signs alternate and the magnitudes decrease.
Step 1: Calculate the First Few Terms
For \(k = 1\): \((-1)^{(1+1)} \cdot \frac{1}{2^1} = (-1)^2 \cdot \frac{1}{2} = \frac{1}{2}\)
For \(k = 2\): \((-1) ^{(2+1)} \cdot \frac{1}{2^2} = (-1)^3 \cdot \frac{1}{4} = -\frac{1}{4}\)
For \(k = 3\): \((-1) ^{(3+1)} \cdot \frac{1}{2^3} = (-1)^4 \cdot \frac{1}{8} = \frac{1}{8}\)
For \(k = 4\): \((-1) ^{(4+1)} \cdot \frac{1}{2^4} = (-1)^5 \cdot \frac{1}{16} = -\frac{1}{16}\)
Continuing this pattern through \(k = 10\), we get:
\(\frac{1}{2}, -\frac{1}{4}, \frac{1}{8}, -\frac{1}{16}, \frac{1}{32}, -\frac{1}{64}, \frac{1}{128}, -\frac{1}{256}, \frac{1}{512}, -\frac{1}{1024}\)
Step 2: Strategic Pairing
Since we have alternating positive and negative terms, let's pair consecutive terms:
Pair 1: \(\frac{1}{2} + (-\frac{1}{4}) = \frac{2}{4} - \frac{1}{4} = \frac{1}{4}\)
Pair 2: \(\frac{1}{8} + (-\frac{1}{16}) = \frac{2}{16} - \frac{1}{16} = \frac{1}{16}\)
Pair 3: \(\frac{1}{32} + (-\frac{1}{64}) = \frac{1}{64}\)
Pair 4: \(\frac{1}{128} + (-\frac{1}{256}) = \frac{1}{256}\)
Pair 5: \(\frac{1}{512} + (-\frac{1}{1024}) = \frac{1}{1024}\)
Step 3: Calculate the Sum
\(T = \frac{1}{4} + \frac{1}{16} + \frac{1}{64} + \frac{1}{256} + \frac{1}{1024}\)
Converting to common denominator 1024:
\(T = \frac{256}{1024} + \frac{64}{1024} + \frac{16}{1024} + \frac{4}{1024} + \frac{1}{1024} = \frac{341}{1024}\)
Step 4: Compare to Answer Choices
Since \(\frac{1}{4} = \frac{256}{1024}\) and \(\frac{1}{2} = \frac{512}{1024}\)
And \(256 < 341 < 512\)
Therefore, \(\frac{1}{4} < T < \frac{1}{2}\)
Answer: D
Want to master the systematic framework for alternating series and discover alternative approaches that work even faster? Check out the complete solution on Neuron by e-GMAT, which reveals the underlying pattern that applies to all similar GMAT problems. Access detailed explanations for official questions with practice quizzes and analytics here.
