Year 12s: what will be the first way you research your future options? >>

Year 12s: what will be the first way you research your future options? >>

Year 12s: what will be the first way you research your future options? >>

Can't even start this question.

I'm looking at this question and I can't even figure out how to start.

I'm not sure what's wrong with me.

Any hints?

From trying the first few values it is obvious that it is going to be less than 3 but I can't really explain why

\displaystyle a_n

must lie in the interval.

(edited 9 years ago)

0057d748f00aba7281fe0d3ff3154d71.png11.0KB

Original post by alex2100x

I'm looking at this question and I can't even figure out how to start.I'm not sure what's wrong with me.Any hints?

From trying the first few values it is obvious that it is going to be less than 3 but I can't really explain why

\displaystyle a_n

must lie in the interval.

Do you know about proof by induction?

OP

Original post by Hodor

Do you know about proof by induction?

Actually I don't get it I have never seen an induction question like this before? Should I rearrange to find a_n in terms of a_n+1????

(edited 9 years ago)

Original post by alex2100x

Actually I don't get it I have never seen an induction question like this before? Should I rearrange to find a_n in terms of a_n+1????

well its super easy!

a_1=5/2

clearly lies between 2 and 3
assume true for n=k and the consider the n=k+1 case using the given formula.

OP

Original post by tombayes

well its super easy!

a_1=5/2

clearly lies between 2 and 3
assume true for n=k and the consider the n=k+1 case using the given formula.

Okay thanks here is what I have got.

Base Case: P(1):

\displaystyle 2 \leq \frac{5}{2} \leq 3

so P(1) is true.

Assume P(k) is true:

2 \displaystyle \leq a_k \leq 3

Show that P(k) being true

\displaystyle \rightarrow

P(k+1) is true:

Here is where I'm stuck by looking at the maximum/minimum values

\displaystyle a_n

can take it seems obvious when substituting for it in

\displaystyle a_{n+1}

that

\displaystyle 2 \leq a_{n+1} \leq 3

but I don't really know how to write this down in a nice way.

OP

I think I've done it.

I showed by solving the inequality for a_n that if a_n is between 2 and 3 and then so much a_n+1 thus if we assume that a_n is in the right interval so must a_n+1 and therefore all natural numbers by induction.

Any ideas how to show it's a decreasing sequence?

Original post by alex2100x

I think I've done it.

I showed by solving the inequality for a_n that if a_n is between 2 and 3 and then so much a_n+1 thus if we assume that a_n is in the right interval so must a_n+1 and therefore all natural numbers by induction.

Any ideas how to show it's a decreasing sequence?

Decreasing basically means that

a_{n+1} < a_n

which is the same as saying that

a_{n+1} - a_n < 0

so a good strategy is usually to try and work with this second inequality to see if you can prove that it's true.

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