# The prize of a lottery is $100 in the first week,$200 in the second week, \$400 in the third week,

#### blamocur

##### Full 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.
There is a typo in the amount for the 9th week.
Also, I assumed that the "accumulated" means the sum of the amounts for each week.

#### eddy2017

##### Senior Member
There is a typo in the amount for the 9th week.
Also, I assumed that the "accumulated" means the sum of the amounts for each week.
Yes, got it.
9 week = 25600
10 week= 51200
Thanks a lot.
Yes, accumulated means that. The sum of the amount for each week which is a series or progression of amounts given, as I studied in sources found on the net when you mentioned the sum of GP. Very interesting and yet uncharted territory for me until yesterday when you mentioned it.

#### eddy2017

##### Senior Member
Formula for sum of geometric progression.
S = a + a(r) + a(r)^2 + . . . + a(r)^n−1
where a = first term
r =ratio
n = the number of terms
In the problem at hand, I am having trouble identifying the ratio? ( maybe a power of 2?).
The ratio being doubled the previous amount, But I do not how to express it. I have studied geometric sequence but the ratio is always the same there( in the tutorial that i watched) but here I have 100 the first week and 200 the second week and then doubles up every week. That throws me off.

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#### eddy2017

##### Senior Member
I think that ratio is common to all the terms in both an arithmetic and a geometric sequence, the difference being that in arithmetic sequence we are dealing with addition and subtraction while in a geometric sequence we are dealing with multiplication and division between terms and that is our case here. But then again, I am failing to recognize the ratio here.
By the way,
Happy thanksgiving to all teachers and tutors on this site!. Big thanks for all of you.

#### lev888

##### Elite Member
Formula for sum of geometric progression.
S = a + a(r) + a(r)^2 + . . . + a(r)^n−1
where a = first term
r =ratio
n = the number of terms
In the problem at hand, I am having trouble identifying the ratio? ( maybe a power of 2?).
The ratio being doubled the previous amount, But I do not how to express it. I have studied geometric sequence but the ratio is always the same there( in the tutorial that i watched) but here I have 100 the first week and 200 the second week and then doubles up every week. That throws me off.
Look up the "direct" formula for the sum.

#### blamocur

##### Full Member
I think that ratio is common to all the terms in both an arithmetic and a geometric sequence, the difference being that in arithmetic sequence we are dealing with addition and subtraction while in a geometric sequence we are dealing with multiplication and division between terms and that is our case here. But then again, I am failing to recognize the ratio here.
By the way,
Happy thanksgiving to all teachers and tutors on this site!. Big thanks for all of you.
I believe they call it "common difference" instead of "ratio" in arithmetic progressions.
As for geometric progression, ratio means the ratio of the [imath]n+1[/imath]th term to the [imath]n[/imath]th term. In your case it would be the ratio of, for examply, 6th week's amount to that of the 5th week.

Happy Thanksgiving everybody, at the very least everybody in the US!

#### eddy2017

##### Senior Member
Look up the "direct" formula for the sum.
Thanks, Mr lev. I am studying it right now. Thanks for the tip.

#### eddy2017

##### Senior Member
I believe they call it "common difference" instead of "ratio" in arithmetic progressions.
As for geometric progression, ratio means the ratio of the [imath]n+1[/imath]th term to the [imath]n[/imath]th term. In your case it would be the ratio of, for examply, 6th week's amount to that of the 5th week.

Happy Thanksgiving everybody, at the very least everybody in the US!
Thanks!

#### eddy2017

##### Senior Member
I just watched and studied a video where a teacher proved how we get to this formula.
I think this is the 'direct' formula' you asked me to look up, Mr lev.
Sn =a(1-r^n) / 1-r

#### eddy2017

##### Senior Member
I believe they call it "common difference" instead of "ratio" in arithmetic progressions.
As for geometric progression, ratio means the ratio of the [imath]n+1[/imath]th term to the [imath]n[/imath]th term. In your case it would be the ratio of, for examply, 6th week's amount to that of the 5th week.

Happy Thanksgiving everybody, at the very least everybody in the US!
Yes, in arithmetic sequence and in geometric sequence they call it 'common difference', but
an arithmetic sequence has a common difference between terms and a geometric sequence has a common ratio between terms.

#### Otis

##### Elite Member
I'm still ahving trouble identifying what the ratio is.
In the formulas, the common ratio is r.

Note that r is also the base of the powers.

What is the base of the powers in your first post?

That is your common ratio. Remember, not all ratios look like fractions.

#### eddy2017

##### Senior Member
The base of powers in my initial post is 2, then, 2 is the common ratio.
Now, my question: is 2 the common ratio because that is the common difference between the terms?. I think the answer is yes.

#### Subhotosh Khan

##### Super Moderator
Staff member
Yes, got it.
9 week = 25600
10 week= 51200
Thanks a lot.
Yes, accumulated means that. The sum of the amount for each week which is a series or progression of amounts given, as I studied in sources found on the net when you mentioned the sum of GP. Very interesting and yet uncharted territory for me until yesterday when you mentioned it.
Not so fast

The accumulated sum is :

S = 100 * (2^10 - 1) =100 * (1024 -1) = 102300

Look up sum of first 'n' terms of a GP series with initial value 'a' and common ratio 'r':

Sn = a * $$\displaystyle \frac{r^n - 1}{r - 1}$$

In your case, n = 10, a=100, & r = 2

Check the answer 102300 using the equation [1]

#### Subhotosh Khan

##### Super Moderator
Staff member
I just watched and studied a video where a teacher proved how we get to this formula.
I think this is the 'direct' formula' you asked me to look up, Mr lev.
Sn =a(1-r^n) / 1-r
You need grouping symbol in the denominator

Sn =a(1- r^n) / (1-r) .......................... when r < 1

Sn =a(r^n - 1) / (r - 1 ) .......................... when r > 1

#### eddy2017

##### Senior Member
Not so fast

The accumulated sum is :

S = 100 * (2^10 - 1) =100 * (1024 -1) = 102300

Look up sum of first 'n' terms of a GP series with initial value 'a' and common ratio 'r':

Sn = a * $$\displaystyle \frac{r^n - 1}{r - 1}$$

In your case, n = 10, a=100, & r = 2

Check the answer 102300 using the equation [1]
I saw your reply Mr Khan. I'm at dinner now so I will check the result with the equation first thing tomorrow morning.
Thanks.
Second equation because r happens to be greater than 1.

#### Subhotosh Khan

##### Super Moderator
Staff member
A brain break is easiest with a friendly woman.

EDIT: Or friendly man under the right circumstances.
Or three other friendly parsons and a deck of cards....

#### Otis

##### Elite Member
Now, my question: is 2 the common ratio because that is the common difference between the terms?. I think the answer is yes.
No, Eddy. You're confusing the geometric sequence with an arithmetic sequence. The terms in an arithmetic sequence differ by the same amount (that's why it's called a common difference).

With geometric sequences, we call r the common ratio because (starting with the second term) the ratio of each term to the prior term is always the same: r. Here are four examples:

800/400 = 2
400/200 = 2
200/100 = 2
an / an-1 = 2

I forgot. Are you working with math textbooks now?

#### eddy2017

##### Senior Member
You need grouping symbol in the denominator

Sn =a(1- r^n) / (1-r) .......................... when r < 1

Sn =a(r^n - 1) / (r - 1 ) .......................... when r > 1
Greetings Mr Khan, I did check it out with the formula and it checks out. The result is the same.
Sn= a(1-r^2) / 1-r
Sn = 100 ( 1-2^n) / (1-2)
Sn =102300