Turning recursive sequence into an explicit sequence

[orchid scroll]

Bunjeet

[flat merlin]

<@&286206848099549185>

<@&286206848099549185>

[flat merlin]

<@&286206848099549185>

[night jacinth]

Can you share the actual question? You should always share the original question, by the way.

[flat merlin]

@night jacinth this is basically the whole question, apart from where the question says to prove the sequence is increasing

i understand to do that i need to do induction

but from what i understand i need to turn this recursive formula into an explicit one in order to be able to do that induction

that is the part i am stuck on

<@&286206848099549185>

[quaint axle]

Just share a picture of the original question

What you think is irrelevant might not be irrelevant to us

[quaint axle]

If you only need to prove that the sequence is increasing then you don’t need an explicit sequence

You can do that with induction

Prove that $x_{n+1}>x_n \Rightarrow x_{n+2}>x_{n+1}$

[orchid scroll]

FirstNameLastName

[flat merlin]

@quaint axle Sorry i dont have the exact question written out properly, just in hand notes, but yes i just have to show that the sequence is increasing and give an upper bound. So, with what you have said here, if i prove x_x+2>xn+1 does it then imply x_n+1 >xn or no?

how would i prove x_n+1>x_n if i dont know what x_n is

thanks for helping aswell

[lethal totem]

If you gonna do this by Induction you don't need to know x_n.

Just begin by showing that for the smallest n = 1 the inequality is true.

Then after you found at least one n ∈ ℕ that makes xₙ₊₁ > xₙ true.
This will be your precondition for your Induction. (1).

∃ n ∈ ℕ : xₙ₊₁ > xₙ (1)

Now the second step is as already mentioned above to show it for n -> n+1.

In other terms:

Prove $x_{n+2}>x_{n+1}$

You need to transform the inequality in such a way where you can use your Induction assumption (1). By doing that you completed the proof then.

[orchid scroll]

adoniszgks

[flat merlin]

thank you

!solved

.solve

.solved