Geometric series

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Ramzi Zeidan
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#1
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#1
Hello,
I really can't distinguish when to use r^n-1 or r^n in various geometric series problems, I do know that r^n-1 is used when we are considering the values at the start of years, and r^n for the values at the end of years, however in mark scheme and questions they use them so differently and I can't understand when to use which.
Help is really appreciated, I am desperate!
Please as quick as possible! Thanks
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gcsemusicsucks
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could you perhaps post an example of where they use it and you don't understand why?
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Ramzi Zeidan
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#3
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(Original post by gcsemusicsucks)
could you perhaps post an example of where they use it and you don't understand why?
6. At the beginning of the year 2000 a company bought a new machine for £15 000. Each year the value of the machine decreases by 20% of its value at the start of the year.

a) Show that at the start of the year 2002, the value of the machine was £9600.
When the value of the machine falls below £500, the company will replace it.
(b) Find the year in which the machine will be replaced.
To plan for a replacement machine, the company pays £1000 at the start of each year into a savings account. The account pays interest at a fixed rate of 5% per annum. The first payment was made when the machine was first bought and the last payment will be made at the start of the year in which the machine is replaced. (c) Using your answer to part (b), find how much the savings account will be worth immediately after the payment at the start of the year in which the machine is replaced.
In part (b) in the mark scheme they used (r)^n why ?

11. Wheat is to be grown on a farm. A model predicts that the mass of wheat harvested on the farm will increase by 1.5% per year, so that the mass of wheat harvested each year forms a geometric sequence. Given that the mass of wheat harvested during year one is 6000 tonnes,
(a) show that, according to the model, the mass of wheat harvested on the farm during year 4 will be approximately 6274 tonnes.
During year N, according to the model, there is predicted to be more than 8000 tonnes of wheat harvested on the farm.
(b) Find the smallest possible value of N
Here is part (b) they used the term r^n-1 but allowed the use of r^n although you get two diffrent values !
Thanks
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Ramzi Zeidan
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#4
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(Original post by Ramzi Zeidan)
6. At the beginning of the year 2000 a company bought a new machine for £15 000. Each year the value of the machine decreases by 20% of its value at the start of the year.

a) Show that at the start of the year 2002, the value of the machine was £9600.
When the value of the machine falls below £500, the company will replace it.
(b) Find the year in which the machine will be replaced.
To plan for a replacement machine, the company pays £1000 at the start of each year into a savings account. The account pays interest at a fixed rate of 5% per annum. The first payment was made when the machine was first bought and the last payment will be made at the start of the year in which the machine is replaced. (c) Using your answer to part (b), find how much the savings account will be worth immediately after the payment at the start of the year in which the machine is replaced.
In part (b) in the mark scheme they used (r)^n why ?

11. Wheat is to be grown on a farm. A model predicts that the mass of wheat harvested on the farm will increase by 1.5% per year, so that the mass of wheat harvested each year forms a geometric sequence. Given that the mass of wheat harvested during year one is 6000 tonnes,
(a) show that, according to the model, the mass of wheat harvested on the farm during year 4 will be approximately 6274 tonnes.
During year N, according to the model, there is predicted to be more than 8000 tonnes of wheat harvested on the farm.
(b) Find the smallest possible value of N
Here is part (b) they used the term r^n-1 but allowed the use of r^n although you get two diffrent values !
Thanks
Please I do need help as fast as possible please!
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Towcestermaths
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#5
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#5
(Original post by Ramzi Zeidan)
Please I do need help as fast as possible please!
In all the formulae, if the value of the first year is a and the ratio is r then the nth value will be ar^n-1 and the sum of the first n terms will be a(1-r^n)/(1-r). What confuses a lot of students is the value that should be used as a.
6a) Assuming that the 15000 value at the beginning of 2000 was your first term in the sequence then a =15000 and r = 0.8. The value at the beginning of 2002 will be the third term and so the value is ar^2 = 9600.
6b) ar^15 =528 and ar^16 = 422. Thus the 16th term is above 500 and the 17th is below. The 16th term is the value at the beginning of 2015 and the 17th term is the value at the beginning of 2016. Hence the machine needed replacing in 2015.
11a) a = 6000 and r = 1.015. Hence the 4th term is ar^3 = 6274
11b) when N=20 the value = ar^19 = 7961. When N=21 the value = ar^20= 8081. Thus the smallest value of N is 21.

In none of these examples should you use r^n, irrespective of the mark scheme comments
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