# 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?

(edited 1 month ago)
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?

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 1 month ago)
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 🙂
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).
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...
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.
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
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.

This is the equation
Original post by KingRich

This is the equation

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
Original post by mqb2766
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.
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.
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