Proving x_n+1 > x_n with Mathematical Induction | Real Number Sequence

[AerospaceEng]

Homework Statement

Consider the sequence of real numbers x₁, x₂, x₃,... defi ned by the relations x₁ = 1

and x_n+1 =[tex]\sqrt{1 + 2x_n}[/tex]

1. Use mathematical induction to show that x_n+1 > x_n
for all n $$\geq$$ 1.

The Attempt at a Solution

I'm a bit thrown off by this question because it seems very obvious that it would be greater. if x₁=1 is it safe to assume that x_n=n? and if so I feel like this proof is stupid cause it shows right in front of you that its greater. any help is great thanks.

 

[Mark44]

Fixed your LaTeX.

Homework Statement

Consider the sequence of real numbers x₁, x₂, x₃,... defi ned by the relations x₁ = 1

and x_n+1 =$$\sqrt{1 + 2x_n}$$

1. Use mathematical induction to show that x_n+1 > x_n
for all n $$\geq$$ 1.

The Attempt at a Solution

I'm a bit thrown off by this question because it seems very obvious that it would be greater. if x₁=1 is it safe to assume that x_n=n? and if so I feel like this proof is stupid cause it shows right in front of you that its greater. any help is great thanks.

The obviousness (or not) is irrelevant to what you need to do, which is to prove this statement by induction.

You are given x₁ = 1. What is the value for x₂? If x₂ > x₁, then use that for your base case.

Next, assume that x_k > x_{k - 1} (the induction hypothesis), and use that assumption to show that x_{k + 1} > x_k. If you can do this, you will have proved that the statement is true for all n >= 1.

 

[╔(σ_σ)╝]

The proof is not stupid! Lol engineers!

Perform the substraction $$x_{n+1}- x_n$$ then show the substraction is > 0. Hint: rationalise and use the fact that a^2 >0 for an real number a.

 

[AerospaceEng]

I'm I allowed to say that x_n+1 is the same thing as x_n + x₁ cause i feel I can't and if I can't I feel so limited.

 

[Char. Limit]

I'm I allowed to say that x_n+1 is the same thing as x_n + x₁ cause i feel I can't and if I can't I feel so limited.

But that's not true.

x_n+1 is clearly shown to be $$\sqrt{1+2x_n}$$.

So...

$$x_{n+1}-x_n = \sqrt{1+2x_n} - x_n$$

So you need to prove that the above equation is always greater than 0, given x₁=1.

 

[AerospaceEng]

okay so I think i see where this is going. what I've done so far is I've taken the RHS and multiplied it by itself just with a positive between the terms, to give me 1+2x_k-x_k²

But i feel like I've done something wrong because i haven't used x₁=1 as of yet..

 

[Char. Limit]

okay so I think i see where this is going. what I've done so far is I've taken the LHS and multiplied it by itself just with a positive between the terms, to give me 1+2x_k+x_k²
and i can factor this to give me (x+1)²
Is that good enough to show that it's always greater than 0. and then ill just plug this mini proof into my main proof.

But i feel like I've done something wrong because i haven't used x₁=1 as of yet..

Considering that x_k itself is always greater than 0, that should be enough. But I'll wait for a second opinion on that.

 

[AerospaceEng]

now if i work with the left hand side I get x_k+1²-x_k² and i can cancel the -x_k² on both sides to give me:

x_k+1²=2_k+1

and i feel I am stuck again and where does the x1=1 come into play?

 

[hunt_mat]

the x_1 comes into play when you show that x_2>x_1 which is the initial step when you should have done first.

 

[Mark44]

now if i work with the left hand side I get x_k+1²-x_k² and i can cancel the -x_k² on both sides to give me:

x_k+1²=2_k+1

and i feel I am stuck again and where does the x1=1 come into play?

See post #2.