Question

Begin with one cent on day 1, four cents on day 2, 16 cents on day three, and so on until the end of 18 days. How much money will there be total?

Answer 1

imran?

Answer 2

Shouldn't that be 18 on top and 1 on the bottom?

Answer 3

16 on 3 imran

Answer 4

And why is it n^2? I thought it would be 4n?

Answer 5

no no neither i believe

Answer 6

not clear is it? it could be
\[4^k\] as well

Answer 7

\[1,4,16,64,...\]

Answer 8

Would choosing this option be more money than 100,000,000?

Answer 9

not
\[2^k\] for sure

Answer 10

n: 1 2 3 4
A: 1 4 16 ?

4^(n-1) right?

Answer 11

i think it is
\[\sum_{k=0}^{18}4^k\]

Answer 12

yes i think satellite is right

Answer 13

you can start at zero and write what i wrote or start at one and write amsitre's

Answer 14

\[\sum_{i = 0}^{17}4^i\]

Answer 15

have we written out a sequence for the partial sums yet? or am i readin ghtis wrong?

Answer 16

anyway it is a huge number because
\[4^{17}=17,179,869,184\]

Answer 17

the total is
\[\frac{4^{18}-1}{4-1}\] whatever that is

Answer 18

Is the total 229064922 by anychance?

Answer 19

22906492245

Answer 20

isnt it just geometric series

Answer 21

yup

Answer 22

there's a formula for this just plug it in

Answer 23

It is "just geometric series", but it's not working out as planned.

Answer 24

I tried that already. I wouldn't be asking if it were that simple.

Answer 25

do you perhapsknow the final answer?

Answer 26

Wait, satellite! Because this is in pennies, wouldn't we have to divide by 100 to convert to dollars?

Answer 27

No hahd, not yet :(

Answer 28

yes you would

Answer 29

Which would then make it LESS than 100,000,000, right?

Answer 30

the formula for summing a geometric series is the one i wrote. it is
\[\frac{a(r^{n+1}-1)}{r-1}\] in this case r is 4 and a is 1 i believe.

Answer 31

A lottery winner must decide between two methods of payment. Choice one is to receive a lump sum of $100,000,000. Choice two is to begin with one cent on day 1, four cents on day 2, 16 cents on day three, and so on until the end of 18 days. Which choice should the winner pick. <-- that is the whole question.

Answer 32

I thought a would be .01 since it is a penny. and why is it n + 1? I thought it was n - 1

Answer 33

Isn't the final answer $229,064,922.45?

Answer 34

end of 18 days maybe means you get that too, so it could be
\[\frac{4^{19}-1}{3}\]
take the series, you will make more for sure

Answer 35

across, that's what I thought, but I think you'd have to divide that by 100 to convert it to dollars because what you have is in pennies.

Answer 36

I already divided it by 100. ^^

Answer 37

give me a second ill writer a program

Answer 38

another form of it is: i think
\[\frac{1-r^n}{1-r}\]

Answer 39

But that can't be. That's the same answer I got without dividing by 100.

Answer 40

http://www.twiddla.com/596084
Let's work it out in here.

Answer 41

22,906,492,245 pennies is what I get

Answer 42

Really? How did you get that? Show me in the whiteboard room.

Answer 43

lets say the amount = S
\[S = 4^0+4^1+4^2+...+4^{n-1}\]now mulitply it all by 4 to get
\(4S =\) \( 4^1+4^2+...+4^{n-1}+4^n\)

subtract them:

S =\( 4^0+4^1+4^2+...+4^{n-1}\)
-4S = \( -4^1-4^2-...-4^{n-1}-4^n\)
------------------------------------
(1-4)S =\(4^0\) \(-4^n\)

now divide off the (1-4) to get

S = \(\cfrac{1-4^n}{1-4}\)

Answer 44

And then just plug in 18 for n and I should get my answer?

Answer 45

yep

Answer 46

thats your answer in pennies :)

Answer 47

Thank you very much amistre!

Answer 48

1 penny = .01 dollars

youre welcome :)