Question

I don't get how to do this one question. :/
A city has a population of 23,000 in 2005, 27,600 in 2006, 33,120 in 2007, and 39,744 in 2008. If this pattern continues, what will be the population in 2012?
Write the iterative rule for the arithmetic or geometric sequence.
Answer the question and show all work.

Answer 1

If it were an arithmetic sequence, there'd be a common difference between each year

If it were a geometric sequence, each successive term would be multiplied by a common ratio

the sequence is 23,000, 27,600, 33,120, 39,744

the first step is finding the ratio (or the difference if it's an arithmetic sequence), then you can either use a formula or count your way to 2012

Answer 2

bibby is right

Answer 3

I didn't say anything rofl

Answer 4

If it's a geometric sequence, the ratio would be 12 correct?
Because 276000/23000 = 12.

Answer 5

if it's a geometric sequence, the ratio would apply to the next 2 terms

Answer 6

so test that out

Answer 7

So it's arithmetic.
39744/33120 =1.2

Answer 8

not exactly. the common ratio is 1.2, you had an extra 0 in the first number which put your ratio off by a factor of 10 :p

Answer 9

Oh, wow.
My bad. So it is geometric.

Answer 10

it is, it is. given a geometric sequence, we can calculate the n-th term using \(a_n=a_1r^{n-1}\)

Answer 11

Ok so a1 would be the first term right?
N is the number of the term, and r would be the ratio?

Answer 12

yep yep yep

Answer 13

an = 23,000(1.2)^12-1

Answer 14

Since it's the 12th in the sequence?

Answer 15

how'd you get 12?
our thingie starts at 2005

Answer 16

So it would be 5-1?
I was thinking backwards ;-;

Answer 17

so if 2005 is the first, we'll do this

Answer 18

\(\large 1_{2005},2_{2006},3_{2007},4_{2008},5_{2009},6_{2010},7_{2011},8_{2012}\)

alternative, I guess we could do 2012-2005+1, but I can't logically explain why that's the case, I just know it, so I'll avoid using it

Answer 19

I mean I guess the + 1 is because we're including 2005. I suck at verbalizing so just use my first statement and count the years

Answer 20

So since there are eight years, it would be...

an = 23,000(1.2)^8-1

Answer 21

a_8 but yeah

Answer 22

What does a_8 mean?

Answer 23

the underscore is LaTeX for a subscript. It means \(a_8\)

Answer 24

Would I still do 8-1?
Orrr would it be something else?

Answer 25

yeah, yeah, sorry for the confusion
recall \(\huge a_n=a_1*r^{n-1}\)

Answer 26

Ok.
an = 23,000(1.2)^8-1
an = 27600^8-1
an =
It gave me some wack decimal. Like, 2.08827...(x10)35
;-;

Answer 27

shh, no tears

PEDMAS/PEMDAS
first, let's do the exponent, 1.2^7=?

Answer 28

3.6

Answer 29

closer to 3.58318. it's better to not round until you finish your calculations cause it's more accurate

23,000*3.583

Answer 30

Oh alright.
I just rounded b/c it's usually what we're told to do.

It would be 82409

Answer 31

we can plug it into wolfram to check
http://www.wolframalpha.com/input/?i=23%2C000%281.2%29%5E7

Answer 32

kinda close

Answer 33

And this would be the population for 2012?

Answer 34

it would. let's check our work another way

Answer 35

\(\begin{array}{|l|c|r|c|l|}
\hline
2005&23,000\\
\hline
2006&27,600\\
\hline
2007&33,120\\
\hline
2008&39,744\\
\hline
2009&?\\
\hline
2010&?\\
\hline
2011&?\\
\hline
2012&?\\
\hline
\end{array}
\)
start at 39,744 and multiply by 1.2 to get 2009
again for 2010
again for 2011
again for 2012

Answer 36

47692
57231
68677
82413

Answer 37

yeaaaaa boiii

Answer 38

Tyyy so much.
And ty for sticking with it :))

Answer 39

it's what I do. thanks for being a good student and actually putting forth the effort. it makes things all the more worthwhile

feel free to tag me in the future @bibby

Answer 40

Will do! Thanks again!