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For n = 1, 2, let Tn = 13 + 23 + ...n3. Which one of the following statements is correct?
- a)There is no value of n for which Tn is a positive power of 2
- b)There is exactly one value of n for which Tn is a positive power of 2
- c)There are exactly two values of n for which Tn is a positive power of 2
- d)There are more than two values of n for which Tn is a positive power of 2
- e)None of these
Correct answer is option 'A'. Can you explain this answer?
Most Upvoted Answer
For n = 1, 2, let Tn = 13 + 23 + ...n3. Which one of the following st...
Sum of 1st n cubes = (n(n+1)/2)2
This cannot be a power of 2 for any positive power of 2 (it can be 20 = 1)
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For n = 1, 2, let Tn = 13 + 23 + ...n3. Which one of the following st...
Understanding Tn
To analyze the sequence Tn = 1^3 + 2^3 + ... + n^3, we first need to derive a formula for Tn. The formula for the sum of cubes of the first n natural numbers is:
- Tn = (n(n + 1) / 2)^2
This means Tn is the square of the sum of the first n natural numbers.
Exploring Tn for Positive Powers of 2
Now, we need to find when Tn could be a positive power of 2, i.e., Tn = 2^k for some integer k.
- Since Tn is a square, it must be in the form of (x^2) where x is an integer.
- Positive powers of 2 that are also perfect squares are limited to 2^0 = 1 and 2^2 = 4, as 2^k is a perfect square only when k is even.
Calculating Tn for n = 1 and 2
Let's calculate Tn for the first few values of n:
- For n = 1:
T1 = 1^3 = 1 (which is 2^0)
- For n = 2:
T2 = 1^3 + 2^3 = 1 + 8 = 9 (not a power of 2)
- For n = 3:
T3 = 1^3 + 2^3 + 3^3 = 1 + 8 + 27 = 36 (not a power of 2)
- Continuing this will show that Tn grows quickly and doesn't yield additional powers of 2.
Conclusion
From these calculations, the only positive power of 2 we found is T1 = 1.
Thus, the correct statement is:
- a) There is no value of n for which Tn is a positive power of 2 (except for T1 which is not considered a natural positive power).
The answer is option A.
To analyze the sequence Tn = 1^3 + 2^3 + ... + n^3, we first need to derive a formula for Tn. The formula for the sum of cubes of the first n natural numbers is:
- Tn = (n(n + 1) / 2)^2
This means Tn is the square of the sum of the first n natural numbers.
Exploring Tn for Positive Powers of 2
Now, we need to find when Tn could be a positive power of 2, i.e., Tn = 2^k for some integer k.
- Since Tn is a square, it must be in the form of (x^2) where x is an integer.
- Positive powers of 2 that are also perfect squares are limited to 2^0 = 1 and 2^2 = 4, as 2^k is a perfect square only when k is even.
Calculating Tn for n = 1 and 2
Let's calculate Tn for the first few values of n:
- For n = 1:
T1 = 1^3 = 1 (which is 2^0)
- For n = 2:
T2 = 1^3 + 2^3 = 1 + 8 = 9 (not a power of 2)
- For n = 3:
T3 = 1^3 + 2^3 + 3^3 = 1 + 8 + 27 = 36 (not a power of 2)
- Continuing this will show that Tn grows quickly and doesn't yield additional powers of 2.
Conclusion
From these calculations, the only positive power of 2 we found is T1 = 1.
Thus, the correct statement is:
- a) There is no value of n for which Tn is a positive power of 2 (except for T1 which is not considered a natural positive power).
The answer is option A.
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For n = 1, 2, let Tn = 13 + 23 + ...n3. Which one of the following statements is correct?a)There is no value of n for which Tn is a positive power of 2b)There is exactly one value of n for which Tn is a positive power of 2c)There are exactly two values of n for which Tn is a positive power of 2d)There are more than two values of n for which Tn is a positive power of 2e)None of theseCorrect answer is option 'A'. Can you explain this answer?
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For n = 1, 2, let Tn = 13 + 23 + ...n3. Which one of the following statements is correct?a)There is no value of n for which Tn is a positive power of 2b)There is exactly one value of n for which Tn is a positive power of 2c)There are exactly two values of n for which Tn is a positive power of 2d)There are more than two values of n for which Tn is a positive power of 2e)None of theseCorrect answer is option 'A'. Can you explain this answer? for CAT 2025 is part of CAT preparation. The Question and answers have been prepared according to the CAT exam syllabus. Information about For n = 1, 2, let Tn = 13 + 23 + ...n3. Which one of the following statements is correct?a)There is no value of n for which Tn is a positive power of 2b)There is exactly one value of n for which Tn is a positive power of 2c)There are exactly two values of n for which Tn is a positive power of 2d)There are more than two values of n for which Tn is a positive power of 2e)None of theseCorrect answer is option 'A'. Can you explain this answer? covers all topics & solutions for CAT 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for For n = 1, 2, let Tn = 13 + 23 + ...n3. Which one of the following statements is correct?a)There is no value of n for which Tn is a positive power of 2b)There is exactly one value of n for which Tn is a positive power of 2c)There are exactly two values of n for which Tn is a positive power of 2d)There are more than two values of n for which Tn is a positive power of 2e)None of theseCorrect answer is option 'A'. Can you explain this answer?.
For n = 1, 2, let Tn = 13 + 23 + ...n3. Which one of the following statements is correct?a)There is no value of n for which Tn is a positive power of 2b)There is exactly one value of n for which Tn is a positive power of 2c)There are exactly two values of n for which Tn is a positive power of 2d)There are more than two values of n for which Tn is a positive power of 2e)None of theseCorrect answer is option 'A'. Can you explain this answer? for CAT 2025 is part of CAT preparation. The Question and answers have been prepared according to the CAT exam syllabus. Information about For n = 1, 2, let Tn = 13 + 23 + ...n3. Which one of the following statements is correct?a)There is no value of n for which Tn is a positive power of 2b)There is exactly one value of n for which Tn is a positive power of 2c)There are exactly two values of n for which Tn is a positive power of 2d)There are more than two values of n for which Tn is a positive power of 2e)None of theseCorrect answer is option 'A'. Can you explain this answer? covers all topics & solutions for CAT 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for For n = 1, 2, let Tn = 13 + 23 + ...n3. Which one of the following statements is correct?a)There is no value of n for which Tn is a positive power of 2b)There is exactly one value of n for which Tn is a positive power of 2c)There are exactly two values of n for which Tn is a positive power of 2d)There are more than two values of n for which Tn is a positive power of 2e)None of theseCorrect answer is option 'A'. Can you explain this answer?.
Solutions for For n = 1, 2, let Tn = 13 + 23 + ...n3. Which one of the following statements is correct?a)There is no value of n for which Tn is a positive power of 2b)There is exactly one value of n for which Tn is a positive power of 2c)There are exactly two values of n for which Tn is a positive power of 2d)There are more than two values of n for which Tn is a positive power of 2e)None of theseCorrect answer is option 'A'. Can you explain this answer? in English & in Hindi are available as part of our courses for CAT.
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For n = 1, 2, let Tn = 13 + 23 + ...n3. Which one of the following statements is correct?a)There is no value of n for which Tn is a positive power of 2b)There is exactly one value of n for which Tn is a positive power of 2c)There are exactly two values of n for which Tn is a positive power of 2d)There are more than two values of n for which Tn is a positive power of 2e)None of theseCorrect answer is option 'A'. Can you explain this answer?, a detailed solution for For n = 1, 2, let Tn = 13 + 23 + ...n3. Which one of the following statements is correct?a)There is no value of n for which Tn is a positive power of 2b)There is exactly one value of n for which Tn is a positive power of 2c)There are exactly two values of n for which Tn is a positive power of 2d)There are more than two values of n for which Tn is a positive power of 2e)None of theseCorrect answer is option 'A'. Can you explain this answer? has been provided alongside types of For n = 1, 2, let Tn = 13 + 23 + ...n3. Which one of the following statements is correct?a)There is no value of n for which Tn is a positive power of 2b)There is exactly one value of n for which Tn is a positive power of 2c)There are exactly two values of n for which Tn is a positive power of 2d)There are more than two values of n for which Tn is a positive power of 2e)None of theseCorrect answer is option 'A'. Can you explain this answer? theory, EduRev gives you an
ample number of questions to practice For n = 1, 2, let Tn = 13 + 23 + ...n3. Which one of the following statements is correct?a)There is no value of n for which Tn is a positive power of 2b)There is exactly one value of n for which Tn is a positive power of 2c)There are exactly two values of n for which Tn is a positive power of 2d)There are more than two values of n for which Tn is a positive power of 2e)None of theseCorrect answer is option 'A'. Can you explain this answer? tests, examples and also practice CAT tests.
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