Question

Find the least integral value of n for the following product to exceed 10^5

Answer 1

\[10^{\frac{ 1 }{ 11 }} \times 10^{\frac{ 2 }{ 11 }} \times 10^{\frac{ 3 }{ 11 }}.......10^{\frac{ n }{11 }}\]

Answer 2

@jim_thompson5910

Answer 3

hmm, well you can add the exponents to get

\[\LARGE 10^{\frac{1}{11}+\frac{2}{11}+\cdots\frac{n}{11}}\]

\[\LARGE 10^{\frac{1+2+\cdots n}{11}}\]

\[\LARGE 10^{\frac{\frac{n(n+1)}{2}}{11}}\]

\[\LARGE 10^{\frac{n(n+1)}{22}}\]

Answer 4

you don't want to exceed 10^5, so the exponent is at most 5

\[\Large \frac{n(n+1)}{22} \le 5\]

Answer 5

solve for n

Answer 6

10 right?

Answer 7

correct

Answer 8

ok thanks

Answer 9

oh sorry you do want to exceed 10^5

n = 10 will make n(n+1)/22 equal to 5

so n > 10 will make n(n+1)/22 more than 5

the next whole number up is 11, so the answer is n = 11

Answer 10

ohhhhhh oh ritght yea