(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

The sequence [tex]u_1,u_2,u_3,...[/tex] is such that [tex]u_1=1[/tex] and [tex]u_{n+1}=-1+{\sqrt{u_n+7}}[/tex]

a) Prove by induction that [tex]u_n<2 for all n\geq1[/tex]

b) show that if [tex]u_n=2-\epsilon[/tex], where [tex]\epsilon[/tex] is small, then [tex]u_{n+1}\approx 2-\frac{1}{6}\epsilon[/tex]

2. Relevant equations

3. The attempt at a solution

[tex]u_{n+1}=-1+sqrt{u_n+7}[/tex]

[tex]\Rightarrow u_n=(u_{n+1}+1)^2-7[/tex]

Assume statement is true for all [tex]k\geq1[/tex]

then [tex]u_k<2[/tex]

[tex]\Rightarrow (u_{k+1}+1)^2-7<2[/tex]

[tex] (u_{k+1}+1)^2-(3)^2<0[/tex]

[tex]((u_{k+1}+1)-3)((u_{k+1}+1)-3)<0[/tex]

[tex](u_{k+1}+1)-3>0[/tex] AND [tex](u_{k+1}+1)-3<0[/tex]

[tex]

u_{k+1}+1>3

u_{k+1}>2

[/tex]

Thus [tex]u_{n+1}>2[/tex] is true

[tex]

(u_{k+1}+1)-3<0

u_{k+1}+1<-3

u_{k+1}<-2

[/tex]

does this affect anything in my proof?

I didn't bother to substitute the values of [tex]u_1[/tex] and [tex]u_2[/tex] and so forth as i have already done it and it is so for all [tex]n\geq1[/tex]

but I do not know how to do part b)

**Physics Forums | Science Articles, Homework Help, Discussion**

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Homework Help: Help in proving sequence by induction

**Physics Forums | Science Articles, Homework Help, Discussion**