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<a https://www.thestudentroom.co.uk/showthread.php?t=7467335'> Year 12s: what will be the first way you research your future options? >>
<a https://www.thestudentroom.co.uk/showthread.php?t=7467335'> Year 12s: what will be the first way you research your future options? >>

Don't understand this sequences question

A pump is used to drain water from a hole on a building site. The pump can drain 10% of the water in the hole during one of its operation cycles. The volume of water in the hole at the start of the pumping process is 120m^3. Show that the total amount of water drained after n cycles is 120(1-0.9^n) m^3
Reply 1
Avatar for mqb2766
mqb2766
21
Why not start by working back from the given answer to understand it?
What is the value when n=0, n=1, n=2, n=3? What do the different parts of the expression represent?
Original post by student638274
A pump is used to drain water from a hole on a building site. The pump can drain 10% of the water in the hole during one of its operation cycles. The volume of water in the hole at the start of the pumping process is 120m^3. Show that the total amount of water drained after n cycles is 120(1-0.9^n) m^3
So when n=0, 0 water is drained, n=1 12m3 is drained, when n=2 22.8m3 is drained. So the equation shows that as n (the no. of cycles) increases, the water drained also increases. But now I don't know what to do?
Original post by mqb2766
Why not start by working back from the given answer to understand it?
What is the value when n=0, n=1, n=2, n=3? What do the different parts of the expression represent?
Reply 3
Avatar for mqb2766
mqb2766
21
It would be less about the final answer, more the bits of the expression
n=0, 120*0
n=1 120*(1-0.9) = 120*0.1
n=2 120*(1-0.81) = 120*0.19

Which part is modelled by a geometric sequence and what is the sequence?

Original post by student638274
So when n=0, 0 water is drained, n=1 12m3 is drained, when n=2 22.8m3 is drained. So the equation shows that as n (the no. of cycles) increases, the water drained also increases. But now I don't know what to do?
I am still confused about the geometric sequence; the sequence is not being multiplied by a constant value of r if I'm not mistaken, or am I missing something?
Reply 5
Avatar for sotor
sotor
14
Original post by student638274
I am still confused about the geometric sequence; the sequence is not being multiplied by a constant value of r if I'm not mistaken, or am I missing something?

if 10% is taken each time, what % of term one is left in term two? and term three?
Reply 6
Avatar for mqb2766
mqb2766
21
Its slightly modified. If you expand the brackets you get

120 - 120*0.9^n

does that help? Draining 10% means 90% is left?

Original post by student638274
I am still confused about the geometric sequence; the sequence is not being multiplied by a constant value of r if I'm not mistaken, or am I missing something?
Would it mean 90% is left in the first one and in the second term aswell?
Original post by sotor
if 10% is taken each time, what % of term one is left in term two? and term three?
Yes it does, so when 10% of water is removed each time there will be 90% left, So to get this equation the initial amount of water is 120 and you subtract the volume of water remaining in the tank which is 120*0.9^n? If it is that how do you form the equation 120*0.9^n?
Original post by mqb2766
Its slightly modified. If you expand the brackets you get

120 - 120*0.9^n

does that help? Draining 10% means 90% is left?
Reply 9
Avatar for mqb2766
mqb2766
21
Sounds good.
Original post by student638274
Yes it does, so when 10% of water is removed each time there will be 90% left, So to get this equation the initial amount of water is 120 and you subtract the volume of water remaining in the tank which is 120*0.9^n? If it is that how do you form the equation 120*0.9^n?
Ah cheers, so this equation 120*0.9^n looks like it is a geometric sequence formula but isn't it layed out as un=ar^(n-1)? Why is it to the power of n in the equation that we formed instead of n-1? Also I was confused because the question seemed like it was asking for the total sum of water removed, which would imply it being a geometric series?
Original post by mqb2766
Sounds good.
0.9^(n-1) = 0.9^n * (1/0.9)
or
0.9^n = 0.9*0.9^(n-1)
They're the same - can both be used to represent the same geometric sequence, but with a slighly different initial values.
Generally, r^n is used when the initial value is at n=0 and r^(n-1) is used when the initial value is at n=1.
Original post by student638274
Ah cheers, so this equation 120*0.9^n looks like it is a geometric sequence formula but isn't it layed out as un=ar^(n-1)? Why is it to the power of n in the equation that we formed instead of n-1? Also I was confused because the question seemed like it was asking for the total sum of water removed, which would imply it being a geometric series?
(edited 4 years ago)
What do you mean by slightly modified initial value? If I was going to prove that equation how would I get to 0.9^n divided by 0.9 because wouldn't that deviate from the initial formula that we are trying to prove 120(1-0.9^n)? Thanks.
Original post by mqb2766
0.9^(n-1) = 0.9^n * (1/0.9)
or
0.9^n = 0.9*0.9^(n-1)
They're the same, but a slighly modified initial value.
Just saw your edit, that makes sense. Just 1 thing to clear up: how do we know that to form this equation we use geometric sequences instead of geometric series?
Original post by mqb2766
0.9^(n-1) = 0.9^n * (1/0.9)
or
0.9^n = 0.9*0.9^(n-1)
They're the same - can both be used to represent the same geometric sequence, but with a slighly different initial values.
Generally, r^n is used when the initial value is at n=0 and r^(n-1) is used when the initial value is at n=1.
I'd slightly modified the previous post. One initial value is 0.9 times or divided by the other so
a*0.9^n = (a*0.9)*0.9^(n-1)
They represent the same sequence, but there is a shift in "n" by one unit hence the "initial" term is shifted by one unit.

Original post by student638274
What do you mean by slightly modified initial value? If I was going to prove that equation how would I get to 0.9^n divided by 0.9 because wouldn't that deviate from the initial formula that we are trying to prove 120(1-0.9^n)? Thanks.
(edited 4 years ago)
A geometric series involves summing (integrating) previous (geometric) values. TBH, you could have done this question as a series where you modelled the amount removed as a sequence, but it would have been more complicated and probably easier to just do a simple sequence and simple algebra?
Original post by student638274
Just saw your edit, that makes sense. Just 1 thing to clear up: how do we know that to form this equation we use geometric sequences instead of geometric series?

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