0
$\begingroup$

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.

$\endgroup$
  • $\begingroup$ 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
0
$\begingroup$

hint

for $n\ge 1$,

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

$$b_n=-100+200n.$$

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

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.