If n is a positive integer such that $$(\sqrt[7]{10})(\sqrt[7]{10})^{2}...(\sqrt[7]{10})^{n}999$$, then th

Question

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

Accepted Answer

(\sqrt[7]{10})(\sqrt[7]{10})^{2}...(\sqrt[7]{10})^{n}&gt;999 (\sqrt[7]{10})^{1+2+...+n}&gt;999 10^{\frac{1+2+...+n}{7}}&gt;999 For minimum value of n, 1 + 2 + . . . + n 7 = 3 \frac{1+2+...+n}{7}=3 7 1 + 2 + ... + n ​ = 3 1 + 2 + ... + n = 21 We can see that if n = 6, 1 + 2 + 3 + ... + 6

If n is a positive integer such that (\sqrt[7]{10})(\sqrt[7]{10})^{2}...(\sqrt[7]{10})^{n}&gt;999, then the smallest value of n is

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