Given the sequences u0=1, un+1=1/3(un+2vn), v0=12, vn+1=1/4(un+3vn), and wn=vn-un, show that (wn) is a…

Question

Given the sequences u0=1, un+1=1/3(un+2vn), v0=12, vn+1=1/4(un+3vn), and wn=vn-un, show that (wn) is a geometric sequence with positive terms.

ScanSolve AI Tutor · Accepted Answer

Here's the solution for question 10.

Given the sequences:
$u_0 = 1$
$u_{n+1} = \frac{1}{3}(u_n + 2v_n)$
$v_0 = 12$
$v_{n+1} = \frac{1}{4}(u_n + 3v_n)$
$w_n = v_n - u_n$

i) Show that $(w_n)$ is a geometric sequence with positive terms.

Express $w_{n+1}$ in terms of $u_n$ and $v_n$ .
$w_{n+1} = v_{n+1} - u_{n+1}$
Substitute the given recurrence relations for $u_{n+1}$ and $v_{n+1}$ :
$w_{n+1} = \frac{1}{4}(u_n + 3v_n) - \frac{1}{3}(u_n + 2v_n)$
Simplify the expression for $w_{n+1}$ .
Find a common denominator, which is 12:
$w_{n+1} = \frac{3(u_n + 3v_n) - 4(u_n + 2v_n)}{12}$ $w_{n+1} = \frac{3u_n + 9v_n - 4u_n - 8v_n}{12}$ $w_{n+1} = \frac{-u_n + v_n}{12}$ $w_{n+1} = \frac{v_n - u_n}{12}$
Relate $w_{n+1}$ to $w_n$ .
Since $w_n = v_n - u_n$ , we have:
$w_{n+1} = \frac{1}{12} w_n$
This shows that $(w_n)$ is a geometric sequence with a common ratio $r = \frac{1}{12}$ .
Check if the terms are positive.
Calculate the first term $w_0$ :

Given the sequences u0=1, un+1=(1)/(3)(un+2vn), v0=12, vn+1=(1)/(4)(un+3vn), and wn=vn−un, show that (wn) is a geometric sequence with positive terms.