We can find the terms of this sequence easily enough.
Here \(b_1\) is just \(a_1\text{,}\) but then
and so on.
There are a few ways we might describe \(b_n\) in general. We could do so recursively as,
since the total number of push-ups done after \(n\) days will be the number done after \(n-1\) days, plus the number done on day \(n\text{.}\)
For something closer to a closed formula, we could write
or the same thing using summation notation:
However, note that these are not really closed formulas since even if we had a formula for \(a_n\text{,}\) we would still have an increasing number of computations to do as \(n\) increases.