Chapter 10: Summing the Natural Numbers: Situation 4 From the MACMTL–CPTM Situations Project

In a mathematical modeling course for prospective secondary mathematics teachers, discussion focused on finding an explicit formula for a sequence expressed recursively. During the discussion, students expressed a need to sum the natural numbers from 1 to n. After several attempts to remember the formula, a student hypothesized that the formula contained n and n + 1. Another student said he thought that it was

but was not sure. During the ensuing discussion, a third student asked, “How do we know that

$n(n+1)2$

is the sum of the integers from 1 to n ? And won't that formula sometimes give a fraction?”