Question

A sequence is defined by f(0) = 3 and f(n) = 2-f(n-1) for n ≥ 1. Which of the following statements defines the nth term of f?

A) f(n) = 3+2 (n-1) for n ≥ 1
B) f(n)= 3 + 2n for n 2 1
C)f(n)=3-2-1 for n ≥ 1
D)f (n)=3-2" for n 21

Answer 1

Hi,

Explanation:

Let's say

`u_0=f(0)\\u_n=2-u_(n-1)\\\\\Longrightarrow\ u_(n+1)=2-u_n\\\\u_0=3\\u_1=2-u_0=2-3=-1\\u_2=2-u_1=2-(-1)=3\\u_3=2-u_2=2-3=-1\\...\\If\ we\ suppose\ that\ the\ sequence\ converge\ to\ x:\\u_(n+1)=2-u_n == > x=2-x == > 2x=2 == > x=1\\Let's\ say\ v_n=u_n-1\\\\v_(n+1)=u_(n+1)-1\\=2-u_n -1\\=1-u_n\\=-(u_n-1)\\=-v_n\\\\v_0=u_0-1=3-1=2\\\\v_n\ is\ a\ geometric\ progression\ with\ v_0=2\ and\ ratio=-1\\\\v_n=2*(-1)^n\\`

`So\ \boxed{u_n=v_n+1=1+2*(-1)^n}\\\\u_0=1+2*(-1)^0=1+2*1=3\\u_1=1+2*(-1)^1=1-2=-1\\u_2=1+2*(-1)^2=1+2=3\\\\u_3=1+2(-1)^3=1-2=-1\\...`