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For a positive integer n, define a(n) = 1 + 1/2 + 1/3 + 1/4 +... + 1/(2ⁿ - 1). Which of the following is true?

1. a(100) is at most 100
2. a(100) is greater than 100
3. a(200) is at most 100
4. a(200) is less than 100

Correct answer: a(100) is at most 100

Solution

a(n)=1+1/2+...+1/(2^n -1) is approximately ln(2^n)=n ln2. So a(100) ~ 69.9 <= 100 (true), while a(200) ~ 138.6 > 100. Hence the correct statement is that a(100) is at most 100.

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