# The prize of a lottery is $100 in the first week,$200 in the second week, $400 in the third week, #### eddy2017 ##### Senior Member The prize of a lottery is$100 in the first week, $200 in the second week,$400 in the third week, and continues to double for a total of 10 weeks. Find the total amount of money that is given in prize.

I found that the expression could be written like this,
total loto prize = 100 * 2^(n-1)
I understand why 100 * 2 (because it says that the prize continue to double)
that n may stand for the number of weeks,
but what I do not understand is why n-1

Any clarification appreciated,
eddy

#### lev888

The prize of a lottery is $100 in the first week,$200 in the second week, $400 in the third week, and continues to double for a total of 10 weeks. Find the total amount of money that is given in prize. I found that the expression could be written like this, total loto prize = 100 * 2^(n-1) I understand why 100 * 2 (because it says that the prize continue to double) that n may stand for the number of weeks, but what I do not understand is why n-1 Any clarification appreciated, eddy The expression you found (looked up?) is the prize for week n. It's not the total prize. What is the math concept that can be used to model this situation? (prize amount doubling every week) #### Otis ##### Elite Member what I do not understand is why n-1 We don't want to double the$100 in the first week, Eddy. That's why we need to reduce the number of weeks by 1.

Try it both ways, and see which gives the correct amount for the first week:

n=1

100*2^(n)

100*2^(n-1)

#### eddy2017

##### Senior Member
100* 2^(n-1)
=100* 2^(10-1)
=100*2^9
=100* 512
=51200 dollars in prize.

#### eddy2017

##### Senior Member
The expression you found (looked up?) is the prize for week n. It's not the total prize.
What is the math concept that can be used to model this situation? (prize amount doubling every week)
Exponential growth because the growth is proportional to the current size of the quantity being discussed.
Now I think I see where this may come from:
The general exponential growth model is:
y=C(1+r)^t

where C is the initial amount or number, r is the growth rate and t is the time elapsed.

#### blamocur

##### Full Member
100* 2^(n-1)
=100* 2^(10-1)
=100*2^9
=100* 512
=51200 dollars in prize.
The answer does not look right to me. Do you know the formula for the sum of geometric progression?

#### lev888

##### Elite Member
Exponential growth because the growth is proportional to the current size of the quantity being discussed.
Now I think I see where this may come from:
The general exponential growth model is:
y=C(1+r)^t

where C is the initial amount or number, r is the growth rate and t is the time elapsed.
Can you apply this formula to the problem?

#### eddy2017

##### Senior Member
Can you apply this formula to the problem?
Allow me to reason this out the easy way and then I will try to apply the formula if this way is correct and please, correct me if I go astray.
I'll try the geometric progression that blamocur mentioned
the initial amount for t he first week ( C) =100
Then, from there onwards the initial amount was doubled every week for ten straight weeks

100+ 200+400+600+800+1000+1,200+ 1,400, + 1,600 + 1,800 = 9100
This is the total amount of the kitty at the end of week 10 if we follow the sum of geometric progression.

#### eddy2017

##### Senior Member
Can you apply this formula to the problem?
Now, Mr Lev, trying to apply the formula of exponential growth
so if C stands for the initial amount then C=100
if r represents the growth rate is doubled every week then I think the r =2 ( I am not sure here).
t the amount of time = 10
if this is correct then the applied formula would be,
y=C(1+r)^t
y= 100( 1+2)^10

Is this okay?.

#### lev888

##### Elite Member
Now, Mr Lev, trying to apply the formula of exponential growth
so if C stands for the initial amount then C=100
if r represents the growth rate is doubled every week then I think the r =2 ( I am not sure here).
t the amount of time = 10
If you are not sure how to use the formula look at a few examples.
Important: this formula gives you the amount at time t. What you need is the total. So you either need to find all individual prizes using the formula above and total them (boring). Or look up the formula for the sum of a geometric sequence.

#### eddy2017

##### Senior Member
I found the formula here. Yes, long involved process, but knowing the formula makes it easy. It is just a question of finding the initial sum ( a) and the growth ratio (r) and substituting them in the formula.

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#### JeffM

##### Elite Member
Allow me to reason this out the easy way and then I will try to apply the formula if this way is correct and please, correct me if I go astray.
I'll try the geometric progression that blamocur mentioned
the initial amount for t he first week ( C) =100
Then, from there onwards the initial amount was doubled every week for ten straight weeks

100+ 200+400+600+800+1000+1,200+ 1,400, + 1,600 + 1,800 = 9100
This is the total amount of the kitty at the end of week 10 if we follow the sum of geometric progression.
But 600 is NOT double 400. 800 is NOT double 600.

#### eddy2017

##### Senior Member
But 600 is NOT double 400. 800 is NOT double 600.

Oh, yes, Mr Jeff, thank for pointing that out. I see now. I'll keep on working tomorrow. I need a brain break.

#### JeffM

##### Elite Member
A brain break is easiest with a friendly woman.

EDIT: Or friendly man under the right circumstances.

#### eddy2017

##### Senior Member
A brain break is easiest with a friendly woman.

EDIT: Or friendly man under the right circumstances.
A friendly woman in my case, lol.

#### lev888

##### Elite Member
A brain break is easiest with a friendly woman.

EDIT: Or friendly man under the right circumstances.
Unless he or she is into brainteasers as a form of relaxation.

#### jonah2.0

##### Full Member
Beer induced ramblings follows.
... I need a brain break.
Put it away and do something else for the rest of the day. Sleep is particularly effective at changing perspective on problems.

Exercise and repetitive, mind numbing activities also helps a lot to accelerate subconscious assimilation. Exercises, especially those exercise that requires you to balance yourself as in chin ups, bench press, and squats or knee bends, does wonders for me. Washing dishes and cleaning something are two mind numbing activities that also works. The fictional scientist Dr. Walter Bishop is said to favor counting electric posts while walking when he feels stuck.

A simple change in seating position is also something to consider. Instead of being all hunched up on your chair, you could lean back on your chair and watch your work from a distance on a black/white board. You could even relax on a lounge chair while you're looking at your board. If you can afford it, you may even want to sit on one of them massage chairs while contemplating your work on the board. Lying down, whilst holding your phone or tablet with your left or right hand as its (your hand) resting against another pillow (or some other convenient object to rest the back of one's hand) is also a good option to consider (It's an option, https://www.freemathhelp.com/forum/goto/post?id=527126, that my twin brother Subhotosh Khan and I are quite fond of). Rene Descartes was allegedly inspired while on his back watching an insect crawling at the ceiling.

You may also consider the using the priority principle, commonly used by bodybuilders in developing a specific muscle group, to break through. As applied to your dilemma, you simply proceed to your study table the moment you wake up when you're all fresh and just had a good sleep (and presumably had some coffee and a meal afterwards).

Bodybuilders usually apply this principle by working out a specific muscle group when they are fresh and could devote all their physical and mental efforts to that specific muscle group like the legs in an exercise session.

Sometimes, as it happens to me, you're working on a problem for a long time without any break that what you need is just a simple bathroom break. But that's just the beer in me bladder talking.

Getting mildly inebriated usually works faster for me though.

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#### eddy2017

##### Senior Member
Unless he or she is into brainteasers as a form of relaxation.
Jjj, that's a good one Mr lev. Lol

#### eddy2017

##### Senior Member
let me see if, after an strenuous workout suggested by jonah, and some teasing, ( not specifically brainteasing [but same effect] rendered by my better half, I manage to get this:
1 week =100
2 week=200
3 week= 400
4 week = 800
5 week=1600
6 week=3200
7 week=6400
8 week=12800
9 week=25000
10 week= 51200
That is the prize money accumulated. If it is not, then neither the workout nor the teasing did me any good. lol
Thanks for always being there for me.
Now, if this is correct and I think it is, i would like to give the sum of a geometric progression a try. let me see if I can.

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#### eddy2017

##### Senior Member
Beer induced ramblings follows.

Put it away and do something else for the rest of the day. Sleep is particularly effective at changing perspective on problems.

Exercise and repetitive, mind numbing activities also helps a lot to accelerate subconscious assimilation. Exercises, especially those exercise that requires you to balance yourself as in chin ups, bench press, and squats or knee bends, does wonders for me. Washing dishes and cleaning something are two mind numbing activities that also works. The fictional scientist Dr. Walter Bishop is said to favor counting electric posts while walking when he feels stuck.

A simple change in seating position is also something to consider. Instead of being all hunched up on your chair, you could lean back on your chair and watch your work from a distance on a black/white board. You could even relax on a lounge chair while you're looking at your board. If you can afford it, you may even want to sit on one of them massage chairs while contemplating your work on the board. Lying down, whilst holding your phone or tablet with your left or right hand as its (your hand) resting against another pillow (or some other convenient object to rest the back of one's hand) is also a good option to consider (It's an option, https://www.freemathhelp.com/forum/goto/post?id=527126, that my twin brother Subhotosh Khan and I are quite fond of). Rene Descartes was allegedly inspired while on his back watching an insect crawling at the ceiling.

You may also consider the using the priority principle, commonly used by bodybuilders in developing a specific muscle group, to break through. As applied to your dilemma, you simply proceed to your study table the moment you wake up when you're all fresh and just had a good sleep (and presumably had some coffee and a meal afterwards).

Bodybuilders usually apply this principle by working out a specific muscle group when they are fresh and could devote all their physical and mental efforts to that specific muscle group like the legs in an exercise session.

Sometimes, as it happens to me, you're working on a problem for a long time without any break that what you need is just a simple bathroom break. But that's just the beer in me bladder talking.

Getting mildly inebriated usually works faster for me though.
Thanks a lot jonah for annotating my post and giving all these good advices. I believe wholeheartedly they are good. I do, myself, a little bit of a workout, if not for muscle volume, at least it helps like you say in relaxing me and clearing my mind. Like you say, lying down on the bench while keeping a little weight over my chest and lookin' at the blue sky of Miami really does wonders. If you are a serious member of this site you need to do that. These amazing guys make your brain work at full speed, jejeje. Thanks. Annotate more often, please!.