a2 sequence math question

these both

1 uses A x R ^n
while the other uses A x R^ n-1? why is that
i thought the equation for geomtreric sequence was A x R^ n-1

It's basically just the difference between calling the first term "term 1" (n=1) and calling it "term 0" (n=0) - it doesn't matter which one you use, as long as you keep track of whichever one you're using and interpret it correctly in context

so for the virus one, why did they do n and not n-1?
would it make a difference if n-1 was used instead
/

They've picked n because they're talking about after n days, so it makes sense to have the second day as n=1, since it is 1 day after the initial day.

If they used n-1 instead, you'd be saying that n=1 on the first day, so you'd follow through to find n similarly to how they have, and then have to subtract 1 to find "after how many days" rather than "on which day". So using the identical method but with n-1 instead gives:

1000 = 100 x (1.04)^(n-1)
10 = (1.04)^(n-1)
(n-1)log(1.04) = 1
n-1 = 58.708
so n = 59.708

This tells you that 1000 are infected on the 60th day, which is 59 days after the 1st day, giving you the 59. This method is equally valid, using n instead of n-1 just saves you having to think through the extra context logic at the end

It's basically just the difference between calling the first term "term 1" (n=1) and calling it "term 0" (n=0) - it doesn't matter which one you use, as long as you keep track of whichever one you're using and interpret it correctly in context

for the answer your meant to round up, so 60 days,
using the other method you get 58.708 days so round up to 59
so a answer of 60 would be wrong no?

so when using A x R ^ n
we say n =0
therefore we don't need to subtract it

Yes, 60 is the answer you get from the numbers and rounding up, but then you need to remember what the context of the question is. The 60 tells you that the 60th day is the first time over 1000 people have the virus, and the question is asking "after how many days do 1000 people have the virus". The 60th day is 59 days after the 1st day, giving you 59.

This is why people use both n and n-1, and choose depending on context. In this case, using n means you don't have to deal with the final subtract 1 at the end, and makes things a little less confusing!

Exactly, calling the first day n=0 means your n value for any other day tells you straight away how many days after the first day it is

so its just context then


so why did they usw the n-1 formula for the second question about the bicycle
we could say day 1 is n=0
and use 10 X 1.1 ^ 25 ?

Yep just context, so if you're fine working out the final adjustments from context then you can stick to just using the one method, but if you struggle with context then using n or n-1 as appropriate might help.

For the second part of the bicycle question, you can use either (there's some logic to using day 1 as n=1 since that lines up with the summation in the first part, but definitely not essential). They've used n-1, meaning day 1 is n=1, so day 25 is n=25, and ar^(n-1) = 10 x 1.1^(25-1) = 10 x 1.1^(24).

If instead you used n, meaning day 1 is n=0, day 25 would be n=24, so you'd still end up with ar^n = 10 x 1.1^(24). Remember that renumbering your first term from 1 to 0 also means all your later terms are renumbered, which is why the calculation is the same whether you use n or n-1.

THIS IS WHAT I NEEDED thank you!! this explanation helps
it doesn't matter which one I use along as I know what day 1 and n equals
so using the virus as a example
If i say day 1 = n=0
100 x 1.04 ^0 = 100
buy if i sau day 1 = n=1
100 x 1.04 ^ 1-1 = 100