Question

PLEASE HELP? I dont think anyone can get this i asked and asked!!! Please try best answer?fan?asap
Describe an infinite geometric series with a beginning value of 2 that converges to 10. What are the first 4 terms of the series?

Answer 1

See the formula for summing the geometric series
http://en.wikipedia.org/wiki/Geometric_series#Formula

when n= infinity and r is less than 1, r^n approaches zero
the sum of an infinite number of terms becomes
\[ S= \sum_{n=0}^{\infty} ar^n= \frac{a}{1-r} \]

Answer 2

a is the first term. they tell you that.
S is the sum of an infinite number of terms. they tell you that.
now use algebra to solve for r.

Answer 3

thats what i dont get it why is there "2 converge to 10" thats what confuses me you dont have to give me the answer but a little more hints would be greatly apprieciated

Answer 4

You have to know about a geometric series
For example, there is this series
It starts with 1/2 as the first term
\[ \frac{1}{2} \]
then we set the next term to 1/2 times the first term. 1/2 * 1/2 is 1/4
\[ \frac{1}{2} + \frac{1}{4}\]
the 3rd term is 1/2 times the second term. 1/2 * 1/4 = 1/8
\[ \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + ...\]

we do that forever

Answer 5

Describe an infinite geometric series with a beginning value of 2 that converges to 10

that is asking for a series whose first term is 2

2
and we multiply the 2 times a factor (which we don't know YET, but will call r)
the first two terms are:
\[ 2 + 2r \]
the 3rd term is r times the 2nd term: r * 2r = 2r^2

\[ 2 + 2r + 2r^2 + ....\]
we keep doing that forever

but when we add them all up, they add up to 10 (the more terms we add the closer we get to 10, but we never *quite* get there) the series converges to 10

Answer 6

meaning i only plug in 2? or 2-10? im sorry i am ust confused about the question but i actually understand evrything your saying just not what the question wants me to do

Answer 7

luckily we have a formula for what the series converges (adds up) to:
\[ 2 + 2r + 2r^2 + 2r^3 + 2r^4 + ....= \frac{2}{1-r} \]
they tell us they want this thing to add up to 10
we can use the formula:
\[ \frac{2}{1-r} = 10 \]
I would "flip" both sides (remember 10 can be thought of as 10/1)
can you do that ?

Answer 8

you want to solve for "r"
there are a few ways to do this.

Answer 9

are you stuck on the algebra ?

Answer 10

would it be r=8?

Answer 11

i multiplied 1-r to 10 which led me to ------ 2= 10 * 1-r.... then i got 2=10-r.
then, -8=-r then divided by -1

Answer 12

no, but you can check your answer this way: replace r with 8 in the equation
\[\frac{2}{1-8} = ?10 \]
1-8 is -7
and 2/-7 is not 10

Answer 13

ill try again

Answer 14

oh, ok,
when you multiply 1-r it is one "chunk" so put it in parens
you get
2 = 10*(1-r)

you have to distribute the 10. (multiply BOTH 1 and -r by 10) But I would not bother
I would first, divide both sides by 10

Answer 15

ill tell you what i just got but it looks backwards.... r= -19

Answer 16

write down your steps, and I'll tell you if you are on the right track

Answer 17

this time i multiplied 10 by 2 nd got ... 1-r= 10*2, then, 1-r=20, subtract 1 and get .....-r=19, then divide and get r= -19

Answer 18

pay attention.
we start at the beginning
\[ \frac{2}{1-r} = 10 \]
multiply both sides by (1-r)
\[ \frac{2}{\cancel{1-r}} \cdot \frac{\cancel{(1-r)}}{1} = 10 \cdot (1-r)\]

Answer 19

now divide both sides by 10. can you do that ?

Answer 20

like this
\[\frac{2}{10} = \cancel{\frac{10}{10}} \cdot (1-r) \]

Answer 21

yeah thats what i got

Answer 22

of course 2/10 is 1/5
\[ \frac{1}{5}= 1-r \]

Answer 23

so i could subtract 1 from1/5 next? to get it isolated?

Answer 24

yes, we could add -1 (or subtract 1, but I always think "ADD" and use negative numbers)
-1+1/5 = 1-1 -r

Answer 25

we get
-1 + 1/5 = -r
it looks nicer if we multiply both sides (ALL TERMS) by -1
-1 * (-1+1/5) = -1*-r
+1 -1/5 = r

Answer 26

4/5?

Answer 27

yes, 4/5
you can check

2/(1-4/5) is 2/(1/5) which is 2*5 = 10

now can you write down the first few terms?
the first one is 2
2
then 2nd term is 2 * 4/5
the third term is the 2nd term times 4/5
and so on

If you are ambitious, you can write down all of them, add them up and see if you get 10 (haha)

Answer 28

btw, try some of Khan's videos
http://www.khanacademy.org/math/algebra/solving-linear-equations-and-inequalities/basic-equation-practice/v/equations-3

because solving equations is important.

Answer 29

@phi thank you so much your the best and i loved how you helped and explain really thank you!!

Answer 30

can i ask one more question? i want to see if i got it right

Answer 31

If you have time, start with this one
http://www.khanacademy.org/math/algebra/solving-linear-equations-and-inequalities/equations_beginner/v/simple-equations

and watch the next few.

Answer 32

?

Answer 33

Justine earned $26,000 during the first year of her job at city hall. After each year she received a 3% raise. Find her total earnings during the first five years on the job. $138,037.53*****
$1,004,704.20
$4,020.51
$108,774.30

am i right?

Answer 34

and i sure will!!

Answer 35

this is a geometric series (what a coincidence).
the starting number is 26000
you multiply by 1.03 ( 0.03 is the raise 1.03 is the salary plus raise)
if we write it out:
26000 + 26000*1.03 + 26000*1.03^2 + 26000*1.03^3+ 26000*1.03^4

we could just add it up, but it's better to use the formula
\[ \sum_{n=0}^{4} 26000 (1.03)^n = 26000 \cdot \frac{1-(1.03)^5}{1-(1.03)}\]

the bottom is 1-1.03 = -.03
the top is 1 - 1.159274 = -0.159274
the sum is
\[ 26000 \cdot \frac{ -0.159274}{-0.03} = 26000*5.309135 = 138037.53

Answer 36

the sum is
\[ 26000 \cdot \frac{ -0.159274}{-0.03} = 26000*5.309135 = 138037.53
\]

Answer 37

so i was right?

Answer 38

I got 138037.53
is that what you picked ?

Answer 39

yeah thats what i picked thats why i put the star and you were right thank you soooooo much:)