i need a quick way to figure out sequences like this

[fast birch]

How do the terms differ?

Can you spot some similarity

[glad bolt]

The uh what is it called the first digits are 3, 13, 23

And 8, 18

Sorry english isn't my first language

[fast birch]

Ok, yep. Now look at some term and at its consecutive term

Can you spot some similarity there?

[glad bolt]

One is plus, one is minus, one is plus

[fast birch]

Yeah

Anything else?

[glad bolt]

All of them are fractions?

[fast birch]

Yeah

Anything else?

Is there something special about their numerator?

Or denominator?

[glad bolt]

They have the common difference of 5 in the bottom

[fast birch]

Exactly

[little cloud]

is this an arithmetico geometric series?

[fast birch]

This will be important, together with this

[fast birch]

And the numerator just stays 1

[glad bolt]

I think the bottom one is arithmetic

The denominator

[fast birch]

Yep

Can you find an explicit form for it?

Atleast for the denominator (bottom) first

[little cloud]

3,8,13,18...

do u mean thhis

[glad bolt]

Yes

Is it

[little cloud]

@glad bolt what exactly do you want to do with this sequnce

[glad bolt]

5x-1

[fast birch]

Does that generate 3?

[glad bolt]

Find the common term

[fast birch]

Or 8?

[glad bolt]

5x-2

[fast birch]

Yeah

[little cloud]

5x-2

okk

noww

[fast birch]

Now how can we generate alternating + and -?

Can you find an expression for that?

[little cloud]

put a (-1)^n

[fast birch]

(I was asking Bill but yes)

[little cloud]

lol

im also here to learn

idk how to solve this either

[fast birch]

Ah, alright

[glad bolt]

Yes (-1)^n

It will make -1,1,-1,1....

[fast birch]

Ok, so [a_1 = (-1)^1 \cdot \frac{1}{5(1) - 2} = - \frac 1 3.] Yep, works. [a_2 = (-1)^2 \cdot \frac{1}{2(5) - 2} = \frac 1 8.] Also works. And so on.

[little cloud]

@glad bolt did you mean general term by common term?

i thought u meant the common difference

or the common ratio

[glad bolt]

Thats what we call it here

Yeah general term ig

[glad bolt]

Thank you for your insight

[glad bolt]

Whats the general term for a series like this

1, -1, 1, -1....

[fast birch]

That's called an "Alternating sequence" if that's what you want to know

[glad bolt]

Yes i think its that

Is it like

(-1)^x-1

[fast birch]

Well, you could have e.g. 1, -4, 9, -16, 25, -36, ...

When you see it's alternating, you know the first factor will be (-1)^n, in this case, if you start with a_1, it will be (-1)^(n + 1) because the first term is positive

And you can continue to just look at the absolute values,

1, 4, 9, 16, 25, 36

And find an explicit form for that.

If you notice that's n^2, then you put it together:

a_n = (-1)^(n + 1) * n^2, for natural n >= 1

[glad bolt]

Wow it is

You guys can math really fast

I think i'll keep practicing till i can get the patterns

Is there really no way to find the general term of a sequence with a calculator?

[fast birch]

It depends on what you did in class, if you haven't done much on this topic, probably the sequences will be rather easy to spot like now

There is an entire theory revolving around finding closed forms of sequences

"recurrence relations"

[glad bolt]

I can definitely find the general term but it takes a long time

Skill issue ig

I really wouldn't have time for trial & error on a 60 min exam with 100 questions

[fast birch]

True. Though in that case, the sequences will probably be rather easy to spot

So yeah, practice and you'll spot stuff easier

[glad bolt]

I'll pray for that too

[fast birch]

Well, you can also practice spotting closed forms for harder sequences, then you won't need to worry..

[fast birch]

You can attempt \begin{enumerate} \item [9, 5, 1, -3, -7, -11, \dots] \item [1, 2, 4, 8, 16, 32, 64, \dots] \item [1, 3, 7, 15, 31, 63, \dots] \item [2, 6, 12, 20, 30, 42, 56, 72, 90, \dots]\end{enumerate}

[rocky token]

Hello!