Question

a1=-4 & a2=12

find the general term an for the geometric sequence

Answer 1

Geometric sequence means you multiplied something to the first term, to get to the second term... so what did you multiply to a1 so that you get a2?

Answer 2

-3

Answer 3

Very good.
So the geometric series has you multiplying -3 at every turn, right?
Starting at at -4
So, the first term is \(-4(-3)^0\) the second term is \(-4(-3)^1\)
Does that give you an idea?

Answer 4

whats next?

Answer 5

Well, you could infer that the third term would be \(-4(-3)^2\) and so on...
It seems the n'th term is given by -4 times -3 raised to n-1

Answer 6

ok now im confused

Answer 7

Well, look at a pattern
\[\large a_1 = -4(-3)^{0}\]\[\large a_2 = -4(-3)^{1}\]\[\large a_3= -4(-3)^{2}\]\[\large a_4 = -4(-3)^{3}\]
and so on...
It's only the exponent of the -3 that changes..

Answer 8

okay so how do i get the answer from that

Answer 9

Well, what's the exponent of -3, when n = 1?

Answer 10

0

Answer 11

and when n=2 ?

Answer 12

1

Answer 13

So you see a pattern? The exponent of -3, when for any value n, is just n-1

Answer 14

so final answer n-1

Answer 15

No... but something related.
Can you now formulate a general rule for
\[\huge a_n = ?\]

Answer 16

-3

Answer 17

Remember, it's just the exponent of -3 that changes.

Answer 18

so an = umm as much as you want lol

Answer 19

Nope. Try again. I even showed you the trend ^ I showed you the values of a1, all the way to a4

Now, find the pattern...

Answer 20

exponents plus 1

Answer 21

No, I mean, give the formula for a_n

Answer 22

i just know that the exponents change depending on what n is

Answer 23

yes... And given a value of n, what would the exponent of -3 be?

Answer 24

0..or 1....or 2

Answer 25

No or's... be sure.

Answer 26

I'm saying since we dont no what n is couldnt it be any

Answer 27

Say, n is some positive integer k... what would be the exponent of -3 then?

Answer 28

I honestly am just getting more frustrated. im sorry

Answer 29

You've got to look beyond the obvious, and into the patterns...
when n = 1, the exponent is zero
when n = 2, the exponent is 1
when n = 3, the exponent is 2
when n = 4, the exponent is 3

So what's the pattern?
when n = k, what would be the exponent?

Answer 30

4

Answer 31

No... that's when n=5
But what about n = k?

Answer 32

ok when n = "k" the exponent would be x

Answer 33

nope

Answer 34

whats k?

Answer 35

A positive integer.

Answer 36

n+1

Answer 37

or minus 1

Answer 38

Well, which is it? It can only be one, you know.

Answer 39

minus

Answer 40

There you go.
Whatever the value of n, you should have noticed that the exponent of -3 was n-1

Answer 41

so now what/?????

Answer 42

So... you now have a general formula for a_n.

Answer 43

n-1 right?

Answer 44

Nope.

Answer 45

this is no help its ok

Answer 46

I'm trying, believe me, but if I try any harder, it'd be just handing over the answer, which none of us really wants :)

Answer 47

run through everything to this point one more time. I don't get it

Answer 48

Okay, First, the pattern...
\[\huge a_1=-4(-3)^{0}\]\[\huge a_2=-4(-3)^{1}\]\[\huge a_3=-4(-3)^{2}\]\[\huge a_4=-4(-3)^{3}\]

Answer 49

Now, it's only the exponent of -3 that changes, right?

Answer 50

you have a very high threshold @terenzreignz ! thats amazing, I'd have been irritated as hell right now had I been at your place :P
nice teaching :)

Answer 51

Thanks. I do try :) @shubhamsrg

Answer 52

right

Answer 53

So, we've worked out that whenever we have a value for n, the exponent of -3 would be...?

Answer 54

n-1

Answer 55

sorry if im "irritating" =(

Answer 56

That's right. Does that not constitute a general formula anymore?
\[\huge a_n =-4(-3)^?\]What should take the place of that question mark?

Answer 57

"That's right" is not a response to "sorry if im "irritating"

Answer 58

n-1

Answer 59

But my question stands...
\[\huge a_n = -4(-3)^?\]

Answer 60

Okay,
\[\huge a_n = -4(-3)^{n-1}\]

Answer 61

Check it. Try it for n = 1, 2, 3, etc...

Answer 62

i completely get it now. thank you

Answer 63

i just was not putting it together

Answer 64

Good.
Just so you know now, -3 here, is what's known as the common ratio, r, of the geometric sequence...
A geometric sequence always takes the form
\[\huge a_n=kr^{n-1}\] Where k is the first term, and r is the common ratio.

Answer 65

Get it? Got it? Good.
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Terence out.