On the approximation by three consecutive continued fractions convergents

Denote by p_n/q_n, n =1,2,3,…pn/qn,n=1,2,3,…, the sequence of continued fraction convergents of the real irrational number x. Define the sequence of approximation coefficients by θ_n := q_n|q_n^x − p_n| ,n= 1,2,3,…θn:=qn|qnx−pn|,n=1,2,3,…. A laborious way of determining the mean value of the sequence |θ_n+1 − θ_n−1 |, n =2,3,…|θ_n+1 − θ_n−1 |, n =2,3,…, is simplified. The method involved also serves for showing that for almost all x the pattern θ_n+1 < θ_n < θ_n-1 occurs with the same asymptotic frequency as the pattern θ_n+1 < θ_n < θ_n−1, namely 0.12109 ^... . All the four other patterns have the same asymptotic frequency 0.18945 ^... . The constants are explicity given.