Question

I am having issues understanding the following question
Find the difference between 2 and the squares of ... (see reply)

Could someone explain how one would start working on a question like this?

Answer 1

\[\frac{1}{1},\frac{3}{2},\frac{7}{5},\frac{17}{12},\frac{41}{29},\frac{99}{70}\]

a) Verify that successive terms of the sequence stand in relation to each other as m/n does to (m + 2n)/(m+n),

b) If m/n is a good approximation of √2 then (m + 2n)/(m+n) is a better one, and that the errors in the two cases are in opposite directions.

c) Find the next three terms of the above sequence

(cit: Glyn James, Modern Engineering Mathematics, exr. 1.2.3.5a-c)

Answer 2

What I figured so far is that, like with any sequences there is a certain pattern going on.

if \[\frac{ 1 }{ 1 } = \frac{ m }{ n }\] then \[ \frac{3}{2} = \frac{m+2n}{m+n}\] and these seem correct. But If id continue deconstructing this I quickly find out that this might not be the case, but it is something different.