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First off, is this your formula?
\[C_t+1=(1-q)C_t+q \gamma\]

yes

thank you for responding!! :)

I found equilibrium which is C^* = 0.2

sorry I'm just laying it all out, trying to understand the question

No not 20%, it was 30% originally

"in his lungs and that he exchanges 30 % of the air in his lungs for ambient air at every breath"

ok, so it does look like it's asking for \(c_4\), but I'm not seeing where you'd put in the 4

ok, so you're supposed to do iterations. that makes sense

Is the 1 supposed to be in the subscript?

Like this?
\[C_1=(1-0.3)(0.2)+(0.3)(0.2)\]
And then use this to find \(C_2\)?

yes, iteration! I think you are on to something... so for C4 - let me see what I get...

so wait shouldn't the 0.2 be 0.1, since the concentration is now 10%?

I meant the 0.3 sorry not the 0.2