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Smallest Sn>5000

I am given the geometric sequence 2,8,32,128..

I am asked to find the smallest n for Sn>5000

I know a=2 , r=4

Using Sn=a(1-r^n)/1-r

We get,
5000< 2(1-4^n)/1-4

I find that n=6.44 to 2. d.p. Nothing states in the question that it must be a whole number..

Is my working out correct?

IMG_0831.jpeg
(edited 2 months ago)
Reply 1
Original post by KingRich
I am given the geometric sequence 2,8,32,128..
I am asked to find the smallest n for Sn>5000
I know a=2 , r=4
Using Sn=a(1-r^n)/1-r
We get,
5000< 2(1-4^n)/1-4
I find that n=6.44 to 2. d.p. Nothing states in the question that it must be a whole number..
Is my working out correct?
IMG_0831.jpeg

Its pretty much implied that its an integer so round up to get the smallest integer satisfying the inequality. The original sequence is defined in terms of n=1,2,3,... and having a term with an index of 2.44 wouldnt make sense.

Note the question is a bit imprecise about whether the first term corresponds to n=0 or n=1 and that obviously affects the answer.

Also in this case its easier to write the sum as
a(r^n-1)/(r-1)
as it keeps things positive. You seem to lose the inequality flips during your working.
(edited 2 months ago)
Reply 2
Original post by mqb2766
Its pretty much implied that its an integer so round up to get the smallest integer satisfying the inequality.
Note the question is a bit imprecise about whether the first term corresponds to n=0 or n=1 and that obviously affects the answer.
Also in this case its easier to write the sum as
a(r^n-1)/(r-1)
as it keeps things positive. You seem to lose the inequality flips during your working.

Crap, of course when I multiply -1 the sign is meant to change direction. I’ll boil it down to me misunderstanding the question then 🙂
Reply 3
Original post by KingRich
Crap, of course when I multiply -1 the sign is meant to change direction. I’ll boil it down to me misunderstanding the question then 🙂

Note what youve posted the question could be either find n (text) or find Sn (image).
Reply 4
Original post by mqb2766
Note what youve posted the question could be either find n (text) or find Sn (image).

What do you mean? Sn(image) as in the Sn=Σ thing?…

I can't recall the actual words but it was something like find the smallest n value for where Sn<5000...
Reply 5
Original post by KingRich
What do you mean? Sn(image) as in the Sn=Σ thing?…
I can't recall the actual words but it was something like find the smallest n value for where Sn<5000...

You could either ask for the value of the index (n or n-1) or the value of the sum. Either/both is fine, but writing "smallest Sn > 500" is different from "smallest n such that Sn > 5000". Not a big deal.
Reply 6
Original post by mqb2766
You could either ask for the value of the index (n or n-1) or the value of the sum. Either/both is fine, but writing "smallest Sn > 500" is different from "smallest n such that Sn > 5000". Not a big deal.

Mmm, I’ll find the question tomorrow and send it tomorrow. Perhaps, it’s another bad structured question. Lol
Reply 7
Original post by mqb2766
You could either ask for the value of the index (n or n-1) or the value of the sum. Either/both is fine, but writing "smallest Sn > 500" is different from "smallest n such that Sn > 5000". Not a big deal.

IMG_0848.jpeg


This is the equation
Reply 8
Original post by KingRich
IMG_0848.jpeg
This is the equation

So youve got the answer?

Note that a duffers solution would simply be to list the values (as r is fairly large so n will be small)
2,8,32,128,512,2048,...
and even without summing them it would be fairly clear that the next one would tip you over the 5000 value
Reply 9
Original post by mqb2766
So youve got the answer?
Note that a duffers solution would simply be to list the values (as r is fairly large so n will be small)
2,8,32,128,512,2048,...
and even without summing them it would be fairly clear that the next one would tip you over the 5000 value

Well, I know r=4, so the solution was is n=7, however, I applied the log to find the closest to be 6.44 without considering that they were asking for a whole number.
Reply 10
Original post by KingRich
Well, I know r=4, so the solution was is n=7, however, I applied the log to find the closest to be 6.44 without considering that they were asking for a whole number.

That wouild be the expected solution to the question, though youd note that n must be a positive integer (original sequence index) so the answer would be 7.
Reply 11
Original post by mqb2766
That wouild be the expected solution to the question, though youd note that n must be a positive integer (original sequence index) so the answer would be 7.

Yeah, I think I was just more confident the fact that I could recall the log approach, so either way, I'm happy. I mean I don't get marked on that question, so it's not an issue

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