WILL MEDAL Can anyone comment the labeled functions for Arithmetic Series, Arithmetic Sequences, Geometric Series, and Geometric Sequences.?

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WILL MEDAL Can anyone comment the labeled functions for Arithmetic Series, Arithmetic Sequences, Geometric Series, and Geometric Sequences.?

Mathematics
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labeled functions?
Like the functions that solve them with a label that shows where everything should go. Say if the question was "Identify the 34th term of the arithmetic sequence 2, 7, 12 .."
I need serious help on this Topic

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\[a_n=a_1+d(n-1) \text{ is an arithemetic sequence with first term } a_1 \\ \text{ and common difference } d \]
just replace d with 7-2 or 12-7 and then replace a_1 with 2 since it is the first term then enter in 34 for n use order of operations to find a_(34)
So it would be \[2_{34}=2_{1}+7-2(34-1)\]
where did you get 2_(34)?
and 2_(1)?
you said replace n with 34
yes but what happen to the a_34?
and why didn't you replace a_1 with 2
you replaced it with 2_1 whatever that means
\[a_n=a_1+d(n-1) \\ a_1 \text{ is the first term } \\ d \text{ is the common difference } \\ a_1 \text{ was given as } 2 \\ \text{ you wanted to know } a_{34} \text{ this is why I said to replace } n \text{ with } 34\]
\[a_{34}=2+5(34-1)\]
7-2 or 12-7 either of these differences will give you the common difference because this an arithmetic sequence 7-2=5 12-7=5 so d=5
anyways just follow order of operations to find a_(34)
Oh okay i see now
What about arithmetic Series.?
here is a sequence of numbers: \[a_1,a_2,a_3,a_4,...,a_n,...\] This is an arithmetic sequence if you have:\[a_1,a_1+d,a_1+2d,a_1+3d,a_1+4d,...,a_1+(n-1)d,... \\ \text{ hope you are seeing that I'm using } \\ a_1=a_1 \\ a_2=a_1+d \\ a_3=a_1+2d \\ a_4=a_1+3d \\ ... \\ a_n=a_1+(n-1)d \text{ or can be written as } a_n=a_1+d(n-1) \\ \] An arithmetic series is just the summing of the terms of an arithmetic sequence. \[a_1,a_2,a_3,a_4,...,a_n,...\] This is a geometric sequence if you have: \[a_1,a_1 r,a_1r^2,a_1r^3,a_1r^4,...,a_1r^{n-1},... \\ \text{ I hope you are seeing that I'm using } \\ a_1=a_1 \\ a_2=a_1r \\ a_3=a_1r^2 \\ a_4=a_1r^3 \\ a_5=a_1r^4 \\ ... \\ a_n=a_1r^{n-1} \] A geometric series is just a summing of the terms of a geometric sequence.
Are you wanting the sum formulas ?
Im still trying to fully comprehend this. I'm not very good at math :/
What is the difference between a series and a sequence.? (Just to add to my notes)
you know what sum means?
A series is a sum of the terms of a sequence.
I was just asking if you knew what sum meant because I basically already said this
Sum is the outcome of adding 2 or more number together.
\[\text{ Sequence of numbers looks like } a_1,a_2,a_3,...,a_n,... \\ \text{ a Series looks like } a_1+a_2+a_3+...+a_n+...\] notice a series is just as I said the sum of the terms of a sequence
you might also have seen this notation for the series: \[\sum_{i=1}^{n}a_i\]
Also a series doesn't always have to start at i=1 and end at n
A series can be infinite. And it can also start at i=2 etc...
Thanks for all your help man, i really appreciate it.
if you want to know the sum formula for an arithmetic series: \[\sum_{i=1}^{n}(a_1+d(i-1)) \\ \sum_{i=1}^{n}a_1 + \sum_{i=1}^{n}di-\sum_{i=1}^{n}d \\ a_1n+d \frac{n(n+1)}{2}-dn \\ \frac{2a_1n}{2}+\frac{dn(n+1)}{2}-\frac{2dn}{2} \\ \frac{2a_1n+dn^2+dn-2dn}{2} \\ \frac{n(2a_1+dn+d-2d)}{2} \\ \frac{n}{2}(2a_1+dn-d) \\ \frac{n}{2}(a_1+a_1+d(n-1)) \\ \frac{n}{2}(a_1+a_n) \\ \frac{n(a_1+a_n)}{2} \\ \text{ so \in conclusion } \\ \sum_{i=1}^{n}(a_i+d(n-1))=\frac{n(a_1+a_n)}{2}\]
all this is saying if you aren't used to sigma notation is that: \[(a_1)+(a_1+d)+(a_1+2d)+\cdots +(a_1+(n-1)d)=\frac{n(a_1+a_n)}{2}\]
\[\sum_{i=1}^{n}a_1 r^{i-1}=a_1 \sum_{i=1}^{n} r^{i-1}=a_1 \frac{r^n-1}{r-1}\]
and that is the sum formula for a geometric series
Okay cool :)
and again if you aren't used to sigma notation that just says: \[(a_1)+(ra_1)+(r^2a_1)+\cdots +(r^na_1)=a_1\frac{r^n-1}{r-1}\]
if the geometric series is infinite though then you have.. \[\sum_{i=1}^{\infty}a_1r^{i-1}=a_1 \frac{1}{1-r} \text{ which only converges for } |r|<1\]
anyways post a few new questions and try to actually practice with these formulas this will still be meaningless to you without some practice

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