given f(4)=-23 and f(9) = -58, for an arithmetic sequence, find the general term. help please!

- anonymous

given f(4)=-23 and f(9) = -58, for an arithmetic sequence, find the general term. help please!

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- katieb

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- Jhannybean

Arithmetic Progression follows the format \[\large a_{n}=a_{1}+(n-1)d\]

- anonymous

Very true. Let's get creative. Normally we work with a1, but let's imagine a4 is the first term and a9 is an. What would n be in that case?

- Jhannybean

Oh? Yeah that's somewhat similar to how I solved it...

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## More answers

- anonymous

Once we know n, we can find d. We can get a1. We can get the general formula.

- anonymous

How far apart are a4 and a9? That amt takes the place of (n-1).

- anonymous

so in this case An would be -58?

- Jhannybean

Oh, i see.

- Jhannybean

5-1...ah.

- anonymous

Yes.

- anonymous

but why does n become 9?

- Jhannybean

No no, it's not 9.

- Jhannybean

I wish d ended up a pretty number. Hmm....

- anonymous

We have to look at what the formula really means. It says:
an = a term
a1 = a term before an
(n-1) = how far apart an and a1 are
d = how far apart consecutive terms are

- anonymous

Okay, so that means its-58= -23+(5-1)d

- anonymous

the question asks to solve for the general term but in this case we find d?

- anonymous

Not 5-1, but acutally 5. See term 4 and term 9 are 5 places apart.

- Jhannybean

\[\large a_{n}=a_{1}+(n-1)d\]
There taking this formula and @mrbarry 's analysis of the arithmetic progression process,
we can say that the distance between f(9) and f(4) is 5 numbers. So 5 = n.
we're given f(4) = -23 so a1 = -23
we're given up to the number we want to find up to (an) and that's represented by f(9) = -58 we're trying to reach -58.

- anonymous

@Jhannybean are you comfortable with my generalization of the formula?

- anonymous

oh gosh, i still dont understand how to solve this question

- Jhannybean

And yes I am @mrbarry

- anonymous

so substitution and elimination are taking place?
-23 = a + d(4-1)
-58 = a + d(9-1)

- anonymous

yup

- anonymous

@talha111 remain calm. This is one of those situations where we have to WORK our way to the solution.
To get from term 4 to term 9 there are five jumps.
To get from -23 to -58 you move -35.
Each jump is worth -35/5 = -7 (Now we have d)
We can go back to the original equation and use term 4 or term 9 for an. We know all but a1 so now we can find it.
a1 =

- anonymous

Arithmetic sequences are linear functions based on a domain of the natural numbers! You are a clever one.

- Jhannybean

Haha. deleted it because I didn't want to confuse her.

- anonymous

Are you with us talha? It's annoying when the teachers talk to each other instead of you.

- anonymous

Im here, its just that ive found a different way to solve the question, using elimination/substitution

- anonymous

which is what ive been doing for the past couple on minutes

- anonymous

Good stuff. What did you find?
You will need to substitute your a1 and d into a generic equation to get the answer we've sought.

- anonymous

so far i found d, which is 2.185

- anonymous

and ive just found a, which is -29.55

- anonymous

d won't be anything but an integer. Check again.

- anonymous

a1 is an integer also.

- anonymous

oh, my bad, ithink ive gotten the general term after fixing it, and thats an=1280(1/2)^n-l

- anonymous

hopefully this is correct

- Jhannybean

i got an interesting way of looking at the formula.|dw:1370489389578:dw| if you didn't understand why (n-1) = 5 and not (9-1)
you have n which is the 9th term you want to reach
then you have n-1 which includes all the preceding terms.
n = 9
n-1 = 4
((n)- (n-1)) =(n-1) = 9 - 4 = 5
Therefore, you replace (n-1) with 5.

- Jhannybean

That's my way of looking at it, heh.

- anonymous

haha yeah i understand that, but the way i learned it was to solve by substitution, so im not sure if got the correct answer

- Jhannybean

You understand that???? AWESOME!!!

- anonymous

-23 = a + d(4-1)
-58 = a + d(9-1)
was a good start.
-23 = a + 3d ---> -1(-23 = a + 3d) ---> 23 = -a -3d
-58 = a + 8d ---> -58 = a + 8d --->-58 = a + 8d
-35 = 5d
-7 = d

- anonymous

yes mr barry!! thats what i got after correcting it!

- anonymous

and then we sub that into the first 1

- anonymous

that as in d=-7

- anonymous

in which with further calculation we get the a value

- Jhannybean

now how do we get a1...

- anonymous

-23 = a + 3d
-23 = a + 3(-7)
-23 = a -21
-2 = a

- anonymous

right on @mrbarry

- anonymous

and thus after further calcs we get an=1280(1/2)^n-1, right?

- anonymous

an = a1 + (n-1)d (sub for a1 and d and simplify to get the general explicit formula)

- anonymous

Whoa. Your an looks like a geometric sequence def.

- Jhannybean

OH nvm i see it now.

- anonymous

wait, after a=-2, what do we do?

- anonymous

Look at my comment just before "Whoa."

- Jhannybean

Lol. @talha111 what you wrote is a geometric sequence, not an arithmetic one.
You're right that it has the form an= a1r^n-1 but we're dealing with an arithmetic sequence.
an = a1+(n-1)d

- anonymous

oh okay, so what we do then is just sub in all the numbers weve gotten

- anonymous

in which im getting -70

- Jhannybean

You found your d = -7, and you needed to find the "general term" which is an, your a1 = -2 so what do you get? :)

- anonymous

OH

- anonymous

5-7n

- Jhannybean

hint: it wont be a number, it'll be an equation.

- anonymous

im so close i can smell it

- anonymous

You got it. an =5- 7n !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!1

- Jhannybean

Ohyesssss

- anonymous

BOOYAH! i need to reward you guys somehow. how do i do so on this site?

- Jhannybean

An = -2 -7(n-1) :) good job.

- anonymous

Wooooooooooooooooooooooooooooo!
WE WIN! Too bad Mr. Math Problem. YOU LOSE!

- Jhannybean

lol..

- anonymous

haha

- anonymous

thankyou guys soo much!

- anonymous

How do I give medals?

- anonymous

no idea, but thanks anyways. i really appreciate it

- Jhannybean

"best response"

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