## sparky16 2 years ago Can someone help with this? :) It is posted below in the comment section so I can use the Equation editor.

1. sparky16

|dw:1362285707700:dw|$\Sigma$

2. sparky16

Find the sum of the arithmetic series! Pleassse help:)

3. hartnn

$$\huge \sum \limits_{k=1}^nk = \dfrac{n(n+1)}{2}$$ and a constant (here, 5) can be taken out of sum sign... :)

4. sparky16

Where r we getting this formula? I'm confused where to start

5. hartnn

thats a standard formula that can be used.

6. hartnn

do you want a formula for sum of terms in arithmetic series ? its equivalent to this formula

7. terenzreignz

Deriving the formula for an arithmetic series?

8. sparky16

Yes that is the formula for the arithmetic series. Where do we go from there?:)

9. sparky16

to find the sum?

10. terenzreignz

Well, maybe best to use this formula instead $\huge \sum \limits_{k=1}^nk = \dfrac{n(n+1)}{2}$ It's simpler :D

11. terenzreignz

So anyway, this sum is just... |dw:1362286375802:dw| right?

12. hartnn

ok, i'll mention the sum of arithmetic series formula also, then you choose which one you wanna use, the series will be 5,10,15,20,.... right ? (with n= 100 terms) he 1st term a1 = 5 and there's a common difference of d=5 then the sum formula is : $$S_n = (n/2)(2a+(n-1)d)$$

13. sparky16

terenzreignz...how do you get these expanded form answer? Can u explain that? sorry

14. sparky16

It is isn't the right answer in the back of the book

15. terenzreignz

That's just kind of the definition. Or you could say the sigma notation is a compact form of the sum. $\huge \sum_{k=1}^n=1+2+3+...+(n-1)+n$ Basically adding all integers starting at 1 to n.

16. terenzreignz

$\huge \sum_{k=1}^nk=1+2+3+...+(n-1)+n$ Bloody typo, sorry :)

17. terenzreignz

So anyway, we can rearrange...|dw:1362286994735:dw|

18. terenzreignz

And notice that these terms...|dw:1362287037927:dw| When added, they all equal n+1

19. sparky16

what is the final answer? the answer is supposed to be 25,250...I just don't know how to get it

20. hartnn

did you try any one of the 2 formula i gave u ??

21. sparky16

Going back to hartnn's response near the beginning. This is like the only formua I'm understanding how to do. $S _{100}= (100(5 + ?))/2$ I'm just confused how to get that number I didn't fill in? It's supposed to be $t _{n}$ (which is last term)

22. sparky16

Anyone?

23. hartnn

oh, you need last term ? the general term is 5k to get last term, just put k=100 in general term so, what u get as last term ?

24. sparky16

So i fill in that question mark spot with 100? Im not sure

25. hartnn

general term =5k with k=100, last term = 5*100 = 500 -_-

26. sparky16

To clarify, this is the formula I am using. $S _{n}=\frac{ n(t _{1}+ t _{n} )}{ 2 }$

27. sparky16

wait how does k =100 again?

28. hartnn

yup, thats the only other i didn't mention :P because you are summing from k=1 (which is 1st term) to k=100 (WHICH IS LAST TERM)

29. hartnn

lower limit givs first term, upper limit gives last term

30. sparky16

ok so 100 is the last term... And why do we multiply it by 5?

31. hartnn

last value of k is 100 general term = 5k so last term = 5*100

32. sparky16

and the difference between the numbers would be five right? or is it something else

33. hartnn

yes, common difference = 2nd term -1st term = 3rd term -2nd term =.... = 10-5 = 15-10 =.... = 5 but thats not required for the formula you are using

34. sparky16

Oh ok that makes sense now. Maybe you should help me on other problems I post. Your a good help!

35. hartnn

ok, sure, but did u get 25250 for this one ?

36. sparky16

Oh wait quick question! What was $t _{1}$ in this equation

37. hartnn

t1 is the 1st term index represents which term

38. hartnn

tn is the n'th term

39. sparky16

the index is which part again?

40. hartnn

index is written as subscript $$\huge t_n$$ here n is index and represent which term, t1 = 1st term, t2 =2nd term, and so on

41. sparky16

so t1 is 5

42. hartnn

yup. as 5 is the 1st term

43. hartnn

any more doubts ?

44. sparky16

ok im sorry so I'm trying to solve it on paper. And I'm confused again.. how did we get $t _{n}$ in the formula

45. sparky16

oh wait my bad! i got it

46. hartnn

you mean in $$S_n = n (t_1+t_n)/2$$ thats a general formula, do you want a derivation of this ?

47. hartnn

oh u got it ? good :)

48. sparky16

What would happen when I solve for $t _{n}$ in an equation where n did not equal 1 and it equaled something else?

49. hartnn

didn't get your question ? do u have specific example for that ? say, if you want to find $$t_5$$, then u put n=5 in general term tn

50. sparky16

|dw:1362289579021:dw| Like this. N does not equal 1

51. hartnn

i think u have made up this question ? or is it from your book/notes ?

52. sparky16

53. sparky16

here Ill give you one from the book. |dw:1362289721775:dw|

54. sparky16

thats 30- m by the way. It kinda got cut off

55. hartnn

so, here your 1st term will start from m=10 in 30-m and last term you'll get by putting m=20 in 30-m

56. sparky16

so how would i get $t _{n}$

57. hartnn

tm = last term, last m =30 so to get last term tm, put m=30 in general term 30-m is this confusing ?

58. sparky16

kinda

59. hartnn

general term tm =30-m fist term t10 = 30-10 (put m=10) last term t20 = 30-20 (put m=20)

60. sparky16

so I would do 30 -10 first , and when do I do the 20? i dont know if that makes sense

61. hartnn

it doesn't -_-

62. sparky16

ok lets not do this problem.. Ill ask my teacher on monday..

63. hartnn

you have formula, you have n, you have 1st term, u have last term....just plug in values! ohh..ok, as u wish

64. sparky16

Ill be posting other problems so u may help me if u want