Hatcher-Topology-of-Numbers · Problem 2.1.3

Continued Fractions

Mathematics — Algebraic Topology · continued-fractions-all-twos

Problem

Let

p_n/q_n

be the value of the continued fraction

\frac{1}{a_1} + \frac{1}{a_2} + \cdots + \frac{1}{a_n}

where each of the

n

terms

a_i

is equal to 2. Thus

p_1/q_1 = 1/2

p_2/q_2 = \frac{1}{2 + \frac{1}{2}} = 2/5

, etc. (a) Find equations expressing

p_n

and

q_n

in terms of

p_{n-1}

and

q_{n-1}

, and use these to write down the values of

p_n/q_n

for

n = 1, 2, 3, 4, 5, 6, 7

. (b) Compute the strip of triangles for

p_7/q_7