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Question 63
If n is a positive integer such that $$(\sqrt[7]{10})(\sqrt[7]{10})^{2}...(\sqrt[7]{10})^{n}>999$$, then the smallest value of n is
Correct Answer: 6
Solution
$$(\sqrt[7]{10})(\sqrt[7]{10})^{2}...(\sqrt[7]{10})^{n}>999$$
$$(\sqrt[7]{10})^{1+2+...+n}>999$$
$$10^{\frac{1+2+...+n}{7}}>999$$
For minimum value of n,
$$\frac{1+2+...+n}{7}=3$$
1 + 2 + ... + n = 21
We can see that if n = 6, 1 + 2 + 3 + ... + 6 = 21.
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