1989 AIME Problems/Problem 1

Problem

Compute $\sqrt{(31)(30)(29)(28)+1}$ .

Solution

Let's call our four consecutive integers $(n-1), n, (n+1), (n+2)$ . Notice that $(n-1)(n)(n+1)(n+2)+1=(n^2+n)^2-2(n^2+n)+1 \Rightarrow (n^2+n-1)^2$ . Thus, $\sqrt{(31)(30)(29)(28)+1} = (29^2+29-1) = 869$

Page

Toolbox

Search

1989 AIME Problems/Problem 1

Problem

Solution

See also