MathDB

Let

\mathcal{H}_n

be the set of all numbers of the form

2 \pm\sqrt{2 \pm\sqrt{2 \pm\ldots\pm\sqrt 2}}

where "root signs" appear

n

times. (a) Prove that all the elements of

\mathcal{H}_n

are real. (b) Computer the product of the elements of

\mathcal{H}_n

. (c) The elements of

\mathcal{H}_{11}

are arranged in a row, and are sorted by size in an ascending order. Find the position in that row, of the elements of

\mathcal{H}_{11}

that corresponds to the following combination of

\pm

signs: \plus{}\plus{}\plus{}\plus{}\plus{}\minus{}\plus{}\plus{}\minus{}\plus{}\minus{}

3