We can find the terms of this sequence easily enough.

\begin{equation*} 3, 8, 14, 24, 33, 33, 45,\ldots\text{.} \end{equation*}

Here \(b_1\) is just \(a_1\text{,}\) but then

\begin{equation*} b_2 = 3+5 = a_1 + a_2\text{,} \end{equation*}
\begin{equation*} b_3 = 3+5+6 = a_1 + a_2 + a_3\text{,} \end{equation*}

and so on.

There are a few ways we might describe \(b_n\) in general. We could do so recursively as,

\begin{equation*} b_n = b_{n-1} + a_n\text{,} \end{equation*}

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

\begin{equation*} b_n = a_1 + a_2 + a_3 + \cdots + a_n\text{,} \end{equation*}

or the same thing using summation notation:

\begin{equation*} b_n = \sum_{i=1}^n a_i\text{.} \end{equation*}

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.