barboat 4 years ago Write the following series by using the sigma notation. 1+8+27+64+...+1000

1. sheg

$1^3 + 2^3 + 3^3 +.....+ 10^3 = \sum_{1}^{10}N^3$

2. FoolForMath

$$\sum \limits_{i=1}^{10} i^3$$

I never Understood sigma notation Someone please Explain Me =)

4. sheg

sigma means summation

5. sheg

see what i have written above i.e., answer

6. FoolForMath

$$\sum$$ This is a capital sigma. It's use is best illustrated by an example: $$\sum_{i = 1}^4 \frac{1}{i} = \frac{1}{1} + \frac{1}{2} + \frac{1}{3} + \frac{1}{4}.$$ You begin by replacing the index (in this case, $i$) with the first value it takes on (iit's lower bound in this case, 1). You then proceed to the next number and keep doing this replacement until you are at the upper limit (in this case, 4). Finally, you add all these terms up.

7. sheg

in above expression it is sum of cubes of first 10 natural numbers

8. sheg

fool is right ^^^^^^

What is there on top ,bottom and left ?

10. FoolForMath

Fool is always right :P :D btw I have to attribute the answer to Austin Mohr read here http://math.stackexchange.com/questions/81921/weird-e-letter-sigma I was too tired to type when there is already a very good explanation ;)

11. sheg

bottom it is the range where bottom is the lowest value that N can have and at top u having the highest value that N can have

12. barboat

Could you explain why you write 10 up there ?

13. FoolForMath

10 is the number of terms or number of iterations

14. sheg

$\sum_{N=1}^{10}N^3 = 1^3 +2^3 + 3^3 + 4^3 +5^3 + 6^3 + 7^3 + 8^3 +9^3 + 10^3$

15. barboat

for this series, 1+3+5+7+..+99 it would be sigma (2n-1). and at the bottom it is r=1, right ? how do i get the top number ?

16. sheg

$\sum_{n= 1}^{50}(2n-1) = 1+3+5+7+....+99$

17. FoolForMath

this is arithmetic progression, how you find the number of terms in an A.P ? ;)

Understood =)

19. sheg

is that ok fool bro

20. barboat

using the formula Sn = n/2(a+l), right ?

21. FoolForMath

I would never dare to doubt sheggy ;)

22. sheg

hahahaha buddy i just asked u

But Sheg why is that 50 at the top?

24. FoolForMath

Yes barboat .. plug in the values of a and b and find n

25. FoolForMath

$\sum_{1}^{99}$ is'nt it like this

Sorry n =1 at the bottom

28. FoolForMath

No, it's an arithmetic progression .. find the n-th term

29. FoolForMath

a = 1 last term = 99 , d = 2 what is n ?

30. sheg

aditi the total number of first 50 odd natural numbers are there so i had put 50 at the top

how come first 50 ?

32. FoolForMath

Yup

34. sheg

and the thing which fool is saying that is also another method which is mostly used............and the thing which i m saying as we are having small size so we can calculate easily

35. FoolForMath

then use it :) and sheggy point of view is also the same .. 1,3,5,7 so what is the 50th odd number ?

i thought the top one was last term and bottom first term

37. FoolForMath

No it is the number of iteration

38. barboat

so, Sn= n/2 (100) ? but what do i write on the Sn side ? i cant solve it otherwise

iteration?

WOW I dont Know Maths =D

41. FoolForMath

number of times you want to execute the operation.

42. sheg

itration amt batao usko

43. FoolForMath

No you know maths .. don't give up so easy :)

bataoo

which one is number of times you want to execute the operation.

46. sheg

hahaha,.............see in simple words it is number of odd numbers that u have to add

47. sheg

aditi in which class u r

49. FoolForMath

number of times you want to execute the operation is upper limit - lower limit..

51. sheg

ok gr8 so u have to work hard

I learnt AP without Sigma

53. FoolForMath

you could write sigma notation in various ways

54. barboat

so, Sn= n/2 (100) ? but what do i write on the Sn side ? i cant solve it otherwise

55. sheg

56. FoolForMath

AP is not sigma .. you will learn sigma probably while doing Riemann sums in definite integral

Ok i understood something =P Thanks Fool and Sheg =D Take My Medals =) here you go

58. FoolForMath

but I knew back from standard VII or VIII during olympaid training and all, however I was never good then :P also no need to buy any book .. just follow OCW it's great resource :)

59. FoolForMath

thanks but I think you did not understand it ? :/

60. sheg

$99 = 1 + (n - 1) \times 2$ $99 -1 = (n - 1) \times 2$ $98 = (n - 1) \times 2$ $\frac{98}{2} = (n - 1)$ $49 = (n - 1)$ $49+1 = n$ $50 = n$

i understood the question barboat asked =)

62. sheg

fool buy books by Dr. K. C. Sinha it will help u alot....the books written by him are simply awesome

books are so heavy , i would rather carry a Laptop =P

64. sheg

@ Barboat what is the formula for nth term of AP $t_{n} = a + (n -1)d$ where $t_{n}$ is the nth term a first term d common difference n number of terms

65. FoolForMath

btw $$1+3+5+7+99 = \sum \limits_{i =-5}^{45} (2n+11)$$ am I right ?

66. sheg

here nth term = 99, a= 1 d = 2 n = ? now plug in these values u will get n

67. FoolForMath

I have read the classics Hall and knight in higher algebra sheggy ;)

68. barboat

isnt the formula Sn=n/2 (2a+(n-1)d) ?

69. sheg

btw indians can also write some classics

70. FoolForMath

sorry buddy I hurt to say but I don't agree . most indian authors plagiarized these classics :(

71. sheg

barboat b4 applying that formula u have to apply nth term formula for finding out number of terms

72. FoolForMath

Do you know abut the famous Kanetkar books for C and datastructure ?

73. sheg

i too carry the same feelings as u but not in case of K. C. Sinha. and in case of finance books i never refer indian authors. I prefer to read other than indian publication house books

74. FoolForMath

I don't know about finance but I haven't found any in my domain ..

75. sheg

yeah i had found in fiance domain but the master of finance field is also from india and whole world is reading his books only and due to whom i had been to this website

76. FoolForMath

Aha that's an interesting fact :)

77. sheg

yeah just google out Aswath Damodaran he is real gem.......m dying to meet this finance wizard

78. FoolForMath

Hmm

79. barboat

mm for the 1+8+27+64+...+1000 series, we can use the Tn=a+(n-1)d formula to find the top number ? but the common difference isnt the same.