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Question 28

Consider the sequence $$2, 2, 4, 4, 4, 4, 6, 6, 6, 6, 6, 6, \ldots$$ Then its 100th term is ______.


Correct Answer: 20

Solution

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The number $$2k$$ is written $$2k$$ times, so after $$k$$ blocks the number of terms written is $$2 + 4 + \cdots + 2k = k(k + 1)$$. For $$k = 9$$ this is 90 and for $$k = 10$$ it is 110, so the 100th term lies in the tenth block. That block consists of the number $$2 \times 10 = 20$$.

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