# inequality between two increasing sequences

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.

• You need to define the sequences not just give the first three terms – Ben W Jan 6 at 19:57
• 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... – Robert Israel Jan 6 at 20:00
• @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 – esc1234 Jan 6 at 20:25

for $$n\ge 1$$,
$$a_n=0.001\times n^2$$
$$b_n=-100+200n.$$