I'm looking to answer the following question:

There are given two increasing sequences $$a_n : 0.001; 0.004; 0.009; ... $$ $$b_n : 100; 300; 500; ... $$

Can the 1st sequence catch up with the 2nd(that is, can the inequality $a_n > b_n$ be satisfied for some $n$)

Looking at the 2 sequences I suspect that $a_n$ will surpass $b_n$ for some n as the intervals for a_n are growing.

How would I describe the situation in mathematical notation? In other words, I can see the answer to the question but I don't know how to write it down mathematically.

  • $\begingroup$ You need to define the sequences not just give the first three terms $\endgroup$ – Ben W Jan 6 at 19:57
  • $\begingroup$ My guess is that $a_n = 10^{-3} n^2$ while $b_n = 100 (2n-1)$. If so, $a_n - b_n$ is a quadratic... $\endgroup$ – Robert Israel Jan 6 at 20:00
  • $\begingroup$ @BenW that was the way it was given in the question and I wasn't sure what you meant by define the sequences but RobertIsrael has given the answer. just define the sequences in that way and then solve for $a_n - b_n > 0$. At least I think that will work $\endgroup$ – esc1234 Jan 6 at 20:25


for $n\ge 1$,

$$a_n=0.001\times n^2$$


  • $\begingroup$ yes, I have it now. Thanks for the tip! $\endgroup$ – esc1234 Jan 6 at 20:27

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