## anonymous one year ago WILL MEDAL Can anyone comment the labeled functions for Arithmetic Series, Arithmetic Sequences, Geometric Series, and Geometric Sequences.?

1. freckles

labeled functions?

2. anonymous

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 .."

3. anonymous

I need serious help on this Topic

4. freckles

$a_n=a_1+d(n-1) \text{ is an arithemetic sequence with first term } a_1 \\ \text{ and common difference } d$

5. freckles

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)

6. anonymous

So it would be $2_{34}=2_{1}+7-2(34-1)$

7. freckles

where did you get 2_(34)?

8. freckles

and 2_(1)?

9. anonymous

you said replace n with 34

10. freckles

yes but what happen to the a_34?

11. freckles

and why didn't you replace a_1 with 2

12. freckles

you replaced it with 2_1 whatever that means

13. freckles

$a_n=a_1+d(n-1) \\ a_1 \text{ is the first term } \\ d \text{ is the common difference } \\ a_1 \text{ was given as } 2 \\ \text{ you wanted to know } a_{34} \text{ this is why I said to replace } n \text{ with } 34$

14. freckles

$a_{34}=2+5(34-1)$

15. freckles

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

16. freckles

anyways just follow order of operations to find a_(34)

17. anonymous

Oh okay i see now

18. anonymous

19. freckles

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.

20. freckles

Are you wanting the sum formulas ?

21. anonymous

Im still trying to fully comprehend this. I'm not very good at math :/

22. anonymous

What is the difference between a series and a sequence.? (Just to add to my notes)

23. freckles

you know what sum means?

24. freckles

A series is a sum of the terms of a sequence.

25. freckles

I was just asking if you knew what sum meant because I basically already said this

26. anonymous

Sum is the outcome of adding 2 or more number together.

27. freckles

$\text{ Sequence of numbers looks like } a_1,a_2,a_3,...,a_n,... \\ \text{ a Series looks like } a_1+a_2+a_3+...+a_n+...$ notice a series is just as I said the sum of the terms of a sequence

28. freckles

you might also have seen this notation for the series: $\sum_{i=1}^{n}a_i$

29. freckles

Also a series doesn't always have to start at i=1 and end at n

30. freckles

A series can be infinite. And it can also start at i=2 etc...

31. anonymous

Thanks for all your help man, i really appreciate it.

32. freckles

if you want to know the sum formula for an arithmetic series: $\sum_{i=1}^{n}(a_1+d(i-1)) \\ \sum_{i=1}^{n}a_1 + \sum_{i=1}^{n}di-\sum_{i=1}^{n}d \\ a_1n+d \frac{n(n+1)}{2}-dn \\ \frac{2a_1n}{2}+\frac{dn(n+1)}{2}-\frac{2dn}{2} \\ \frac{2a_1n+dn^2+dn-2dn}{2} \\ \frac{n(2a_1+dn+d-2d)}{2} \\ \frac{n}{2}(2a_1+dn-d) \\ \frac{n}{2}(a_1+a_1+d(n-1)) \\ \frac{n}{2}(a_1+a_n) \\ \frac{n(a_1+a_n)}{2} \\ \text{ so \in conclusion } \\ \sum_{i=1}^{n}(a_i+d(n-1))=\frac{n(a_1+a_n)}{2}$

33. freckles

all this is saying if you aren't used to sigma notation is that: $(a_1)+(a_1+d)+(a_1+2d)+\cdots +(a_1+(n-1)d)=\frac{n(a_1+a_n)}{2}$

34. freckles

$\sum_{i=1}^{n}a_1 r^{i-1}=a_1 \sum_{i=1}^{n} r^{i-1}=a_1 \frac{r^n-1}{r-1}$

35. freckles

and that is the sum formula for a geometric series

36. anonymous

Okay cool :)

37. freckles

and again if you aren't used to sigma notation that just says: $(a_1)+(ra_1)+(r^2a_1)+\cdots +(r^na_1)=a_1\frac{r^n-1}{r-1}$

38. freckles

if the geometric series is infinite though then you have.. $\sum_{i=1}^{\infty}a_1r^{i-1}=a_1 \frac{1}{1-r} \text{ which only converges for } |r|<1$

39. freckles

anyways post a few new questions and try to actually practice with these formulas this will still be meaningless to you without some practice