anonymous
  • anonymous
given f(4)=-23 and f(9) = -58, for an arithmetic sequence, find the general term. help please!
Mathematics
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SOLVED
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katieb
  • katieb
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Jhannybean
  • Jhannybean
Arithmetic Progression follows the format \[\large a_{n}=a_{1}+(n-1)d\]
anonymous
  • 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
  • Jhannybean
Oh? Yeah that's somewhat similar to how I solved it...

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anonymous
  • anonymous
Once we know n, we can find d. We can get a1. We can get the general formula.
anonymous
  • anonymous
How far apart are a4 and a9? That amt takes the place of (n-1).
anonymous
  • anonymous
so in this case An would be -58?
Jhannybean
  • Jhannybean
Oh, i see.
Jhannybean
  • Jhannybean
5-1...ah.
anonymous
  • anonymous
Yes.
anonymous
  • anonymous
but why does n become 9?
Jhannybean
  • Jhannybean
No no, it's not 9.
Jhannybean
  • Jhannybean
I wish d ended up a pretty number. Hmm....
anonymous
  • 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
  • anonymous
Okay, so that means its-58= -23+(5-1)d
anonymous
  • anonymous
the question asks to solve for the general term but in this case we find d?
anonymous
  • anonymous
Not 5-1, but acutally 5. See term 4 and term 9 are 5 places apart.
Jhannybean
  • 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
  • anonymous
@Jhannybean are you comfortable with my generalization of the formula?
anonymous
  • anonymous
oh gosh, i still dont understand how to solve this question
Jhannybean
  • Jhannybean
And yes I am @mrbarry
anonymous
  • anonymous
so substitution and elimination are taking place? -23 = a + d(4-1) -58 = a + d(9-1)
anonymous
  • anonymous
yup
anonymous
  • 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
  • anonymous
Arithmetic sequences are linear functions based on a domain of the natural numbers! You are a clever one.
Jhannybean
  • Jhannybean
Haha. deleted it because I didn't want to confuse her.
anonymous
  • anonymous
Are you with us talha? It's annoying when the teachers talk to each other instead of you.
anonymous
  • anonymous
Im here, its just that ive found a different way to solve the question, using elimination/substitution
anonymous
  • anonymous
which is what ive been doing for the past couple on minutes
anonymous
  • 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
  • anonymous
so far i found d, which is 2.185
anonymous
  • anonymous
and ive just found a, which is -29.55
anonymous
  • anonymous
d won't be anything but an integer. Check again.
anonymous
  • anonymous
a1 is an integer also.
anonymous
  • anonymous
oh, my bad, ithink ive gotten the general term after fixing it, and thats an=1280(1/2)^n-l
anonymous
  • anonymous
hopefully this is correct
Jhannybean
  • 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
  • Jhannybean
That's my way of looking at it, heh.
anonymous
  • 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
  • Jhannybean
You understand that???? AWESOME!!!
anonymous
  • 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
  • anonymous
yes mr barry!! thats what i got after correcting it!
anonymous
  • anonymous
and then we sub that into the first 1
anonymous
  • anonymous
that as in d=-7
anonymous
  • anonymous
in which with further calculation we get the a value
Jhannybean
  • Jhannybean
now how do we get a1...
anonymous
  • anonymous
-23 = a + 3d -23 = a + 3(-7) -23 = a -21 -2 = a
anonymous
  • anonymous
right on @mrbarry
anonymous
  • anonymous
and thus after further calcs we get an=1280(1/2)^n-1, right?
anonymous
  • anonymous
an = a1 + (n-1)d (sub for a1 and d and simplify to get the general explicit formula)
anonymous
  • anonymous
Whoa. Your an looks like a geometric sequence def.
Jhannybean
  • Jhannybean
OH nvm i see it now.
anonymous
  • anonymous
wait, after a=-2, what do we do?
anonymous
  • anonymous
Look at my comment just before "Whoa."
Jhannybean
  • 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
  • anonymous
oh okay, so what we do then is just sub in all the numbers weve gotten
anonymous
  • anonymous
in which im getting -70
Jhannybean
  • 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
  • anonymous
OH
anonymous
  • anonymous
5-7n
Jhannybean
  • Jhannybean
hint: it wont be a number, it'll be an equation.
anonymous
  • anonymous
im so close i can smell it
anonymous
  • anonymous
You got it. an =5- 7n !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!1
Jhannybean
  • Jhannybean
Ohyesssss
anonymous
  • anonymous
BOOYAH! i need to reward you guys somehow. how do i do so on this site?
Jhannybean
  • Jhannybean
An = -2 -7(n-1) :) good job.
anonymous
  • anonymous
Wooooooooooooooooooooooooooooo! WE WIN! Too bad Mr. Math Problem. YOU LOSE!
Jhannybean
  • Jhannybean
lol..
anonymous
  • anonymous
haha
anonymous
  • anonymous
thankyou guys soo much!
anonymous
  • anonymous
How do I give medals?
anonymous
  • anonymous
no idea, but thanks anyways. i really appreciate it
Jhannybean
  • Jhannybean
"best response"

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