cat 2021 Question 20

Question

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

Accepted Answer

$(\sqrt[7]{10})(\sqrt[7]{10})^2 \ldots (\sqrt[7]{10})^n \gt 999$

$(\sqrt[7]{10})^{1+2+\ldots+n} \gt 999$

$10^{\frac{1+2+\ldots+n}{7}} \gt 999$

For minimum value of $n$,

$\frac{1+2+\ldots+n}{7} = 3$

$1 + 2 + \ldots + n = 21$

We can see that if $n = 6$, $1 + 2 + 3 + \ldots + 6 = 21$.

Question 20.