Which of the following correctly describes the nature of the number \(0.101001000100001\ldots\)?
Answer and explanation
Correct answer: Irrational number
The decimal expansion is infinite and the gaps of zeros between successive 1s increase: patterns of 1 appear at positions 1, 3, 6, 10, ... (triangular numbers). A rational number must have a decimal expansion that eventually becomes periodic (a fixed repeating block). Since no such periodic block exists here, the number is irrational. The closest distractor, rational number, is incorrect because there is no eventual repetition; terminating decimal is wrong because the expansion does not end; integer is impossible because the value lies between 0 and 1. Exam tip: to test for irrationality, look for eventual periodicity — if none exists and the expansion is infinite, the number is irrational.
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What is the correct answer to this question?
Irrational number
Why is this the correct answer?
The decimal expansion is infinite and the gaps of zeros between successive 1s increase: patterns of 1 appear at positions 1, 3, 6, 10, ... (triangular numbers). A rational number must have a decimal expansion that eventually becomes periodic (a fixed repeating block). Since no such periodic block exists here, the number is irrational. The closest distractor, rational number, is incorrect because there is no eventual repetition; terminating decimal is wrong because the expansion does not end; integer is impossible because the value lies between 0 and 1. Exam tip: to test for irrationality, look for eventual periodicity — if none exists and the expansion is infinite, the number is irrational.
Which subject and chapter does this question cover?
This is a Class 10 Mathematics question. Chapter: Polynomials. Topic: Irrational numbers and real numbers.
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