Solution 2.2.4.1.

The idea is to mimic how we found the formula for triangular numbers. If we add the first and last terms, we get 472. The second term and second-to-last term also add up to 472. To keep track of everything, we might express this as follows. Call the sum \(S\text{.}\) Then,

\(S =\)	\(2\)	\(+\)	\(5\)	\(+\)	\(8\)	\(+ \cdots +\)	\(467\)	\(+\)	470
\(+ \quad S =\)	\(470\)	\(+\)	\(467\)	\(+\)	\(464\)	\(+ \cdots +\)	\(5\)	\(+\)	2
\(2S =\)	\(472\)	\(+\)	\(472\)	\(+\)	\(472\)	\(+ \cdots +\)	\(472\)	\(+\)	\(472\)

To find \(2S\) then we add 472 to itself a number of times. What number? We need to decide how many terms (summands) are in the sum. Since the terms form an arithmetic sequence, the \(n\)th term in the sum (counting \(2\) as the 0th term) can be expressed as \(2 + 3n\text{.}\) If \(2 + 3n = 470\) then \(n = 156\text{.}\) So \(n\) ranges from 0 to 156, giving 157 terms in the sum. This is the number of 472's in the sum for \(2S\text{.}\) Thus

\begin{equation*} 2S = 157\cdot 472 = 74104\text{.} \end{equation*}

It is now easy to find \(S\text{:}\)

\begin{equation*} S = 74104/2 = 37052\text{.} \end{equation*}