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C2[Edexcel] - Geometric sequence & series

A liquid is kept in a barrel. At the start of a year the barrel is filled with 160 liters of the liquid. Due to evaporation, at the end of every year the amount of liquid in the barrel is reduced by 15% of its volume at the start of the year.

At the start of each year a new barrel in filled with 160 liters of liquid so that, at the end of 20 years, there are 20 barrels containing liquid.

Calculate the total amount of liquid, to the nearest liter, in the barrels at the end of 20 years.
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So far I thought,

The sequence for 20 years will be:
160 +[160(0.85) + 160] + [160(0.85)^2 + 160(0.85) + 160] + … + [160(0.85)^19 + 160(0.85)^18 + … + 160]

So it’s like to calculate each of term.
How could I calculate easily, using formula?

EDIT: Sorry for not mentioning exam board.

Reply 1

deadgenius

geton

So it’s like to calculate each of term.
How could I calculate easily, using formula?

You are right, but there is a much easier way to calculate this.

You are basically looking for the sum of all the values,

S_n

, and depending on which exam board you use, it may have the formula for this in the formula booklet. However if it isn't in the booklet you need to learn this formula:

S_n = \frac{a(1-r^n)}{1-r}

Where a= the starting value, in this case 160,

r= the relationship between each value, in this case 0.85,

n= the number of repititions, in this case 20.

So for this question,

S_n = \frac{160(1-0.85^{20})}{1-0.85}

If you still don't get it, either quote me or PM me and I'll get back to you.

Reply 2

geton

deadgenius

You are right, but there is a much easier way to calculate this.

You are basically looking for the sum of all the values,

, and depending on which exam board you use, it may have the formula for this in the formula booklet. However if it isn't in the booklet you need to learn this formula:

Where a= the starting value, in this case 160,

r= the relationship between each value, in this case 0.85,

n= the number of repititions, in this case 20.

So for this question,

Well,

= 1025.32 (2 d.p.)

But book's answer is 872.

In this sequence,

160 + [160(0.85)+160] + [160(0.85)^2 + 160(0.85) + 160] + ... + [160(0.85)^{19} + 160(0.85)^{18} + ... + 160],

each one seems a sequence. May be we've to think more about this.

Reply 3

deadgenius

geton

In this sequence,

160 + [160(0.85)+160] + [160(0.85)^2 + 160(0.85) + 160] + ... + [160(0.85)^{19} + 160(0.85)^{18} + ... + 160],

each one seems a sequence. May be we've to think more about this.

You are wrong. I thought you were right, but I overlooked something, sorry! it says in the question that it wants the amount in the 20 barrels after 20 years. In your sequence, you are adding up the amount in all 20 barrels after 20 years + amount in all 19 barrels after 19 years + amount in all 18 barrels after 18 years.... etc etc, when in fact you only need to add up the amount in all 20 barrels after 20 years.

So, I was right in which formula I suggested, but I explained it the wrong way. I did the calculations in the normal and very slow way by adding 160 + 160x0.85 + 160 X0.85^2 +.... + 160x0.85^19 and I got over 1000 (I can't remember exactly what), so I think maybe the book is wrong. Hopefully someone else might look at this and sort it all out for us. I really shouldn't be struggling with this!

Reply 4

Soya15

I'm sure the book is wrong. Sometimes the answers in my mechanics textbook are different to what our teacher and our class got. So don't worry.

Reply 5

abrarda1

Ok this is 2 years late but i worked out the answer. Basically you use the formula but for a you use 160*0.85 rather than 160. This gives you 136* (0.85^20 -1)/ (0.85-1). This gives 871.52 litres. rounded = 872litres. :smile:

Reply 6

Theafricanlegend

ahaaha im now two years late, but hopefully you can still help me!! why did you multiply 160 by 0.85???

thannnnnnnnnnnnnnnnnnnks xxoooxoxox :confused:

Reply 7

CharlieBoardman

160 litres is reduced by 15% each year?

Reply 8

LolaMac

Original post by Theafricanlegend

ahaaha im now two years late, but hopefully you can still help me!! why did you multiply 160 by 0.85???

thannnnnnnnnnnnnnnnnnnks xxoooxoxox :confused:

I'm 2 years late too, it's because by the end of that same year 15% of water will be lost through evaporation