1. Addition of Sequences of Numbers: First Principles
= …+ (N-1) + N = ?
= …+ N2 = ?
 n2
n = 1 N
= …+ N3 = ?
 n3
n = 1 N
Derivation based on: R. N. Zare, Angular Momentum, Chap. 1, (1988)

2. Addition of Sequences of Numbers: First Principles
(n+1) – n =
n = 1 N …
+ N +(N-1) + N+1-N

3. Addition of Sequences of Numbers: First Principles
(n+1) – n =
n = 1 N …
+ N +(N-1) + N+1-N

4. Addition of Sequences of Numbers: First Principles
(n+1) – n =
n = 1 N …
+ N +(N-1) + N+1-N

5. Addition of Sequences of Numbers: First Principles
This will disappear with the next term
(n+1) – n =
n = 1 N …
+ N +(N-1) + N+1-N

6. Addition of Sequences of Numbers: First Principles
This will disappear with the next term
(n+1) – n =
n = 1 N …
+ N +(N-1) + N+1-N
This disappears with the previous term

7. Addition of Sequences of Numbers: First Principles
(n+1) – n =
n = 1 N …
+ N +(N-1) + N+1-N
This will disappear with the next term
This disappears with the previous term

8. Addition of Sequences of Numbers: First Principles
(n+1) – n =
n = 1 N …
+ N +(N-1) + N+1-N
The surviving terms

9. Addition of Sequences of Numbers: First Principles
(n+1) – n =
n = 1 N
N+1-1
= N
(n+1) – n =
n = 1 N
n = 1 N
 1 = …+1+1
N1, or

10. Addition of Sequences of Numbers: First Principles
(n+1) – n =
n = 1 N
N+1-1
= N
(n+1) – n =
n = 1 N
n = 1 N
 1 = …+1+1
N1, or N

11. Addition of Sequences of Numbers: First Principles
(n+1)2 – n2 =  N
n=1
(N+1)2 - 12
The last: n = N+1 &
The first n = 1
The surviving terms:
All other terms cancel out

12. Addition of Sequences of Numbers: First Principles
(n+1)2 – n2 =  N
n=1
(N+1)2 - 12
= N2 + 2N + 1 – 1
= N2 + 2N + 1 – 1
= N2 + 2N = N(N+2)

13. Addition of Sequences of Numbers: First Principles
(n+1)2 – n2 =  N
n=1
N(N+2)
n2 + 2n + 1 – n2
n2 + 2n + 1 – n2
(2n + 1) =  N
n=1
N(N+2)

14. Addition of Sequences of Numbers: First Principles N
n=1
N(N+2)
2n +  N
n=1  N
n=1 1 N

15. Addition of Sequences of Numbers: First Principles N
n=1
N(N+2)  N
n=1
2n + N
= N2 + 2N

16. Addition of Sequences of Numbers: First Principles N
n=1
N(N+2)  N
n=1 2n
= N2 + N
= N(N+1) n  N
n=1
= N(N+1) 2

17. Addition of Sequences of Numbers: First Principles
n2 ,
To determine
first determine 
(n+1)3 – n3

18. Addition of Sequences of Numbers: First Principles
(n+1)3 – n3 =  N
n=1
(N+1)3 - 13
The last: n = N+1 &
The first n = 1
The surviving terms:
All other terms cancel out

19. Addition of Sequences of Numbers: First Principles
(n+1)3 – n3 =  N
n=1
(N+1)3 - 13
= N3 + 3N2 + 3N + 1 – 1
= N3 + 3N2 + 3N

20. Addition of Sequences of Numbers: First Principles
(n+1)3 – n3 =  N
n=1
N3 + 3N2 + 3N
n3 + 3n2 + 3n + 1 – n3
n3 + 3n2 + 3n + 1 – n3
(3n2 + 3n + 1) =  N
n=1
N3 + 3N2 + 3N

21. Addition of Sequences of Numbers: First Principles
(3n2 + 3n + 1) =  N
n=1
N3 + 3N2 + 3N
3N(N+1) 2 N
3N(N+1)

22. Addition of Sequences of Numbers: First Principles N
n=1
3N(N+1) 2
+ N
= N3 + 3N(N+1)

23. Addition of Sequences of Numbers: First Principles N
n=1 N
= N3 +
3N(N+1) 2
= N3 + 3N2 + 3N 2

24. Addition of Sequences of Numbers: First Principles N
n=1 N
= N3 + 3N2 + 3N 2

25. Addition of Sequences of Numbers: First Principles N
n=1
= N3 + 3N2 + N 2 n2  N
n=1
= N3 + N N 2 6 3

26. Addition of Sequences of Numbers: First Principles N
n=1
= N3 + N N 2 6 3
= N(N+1)(2N+1) 6

27. Addition of Sequences of Numbers: First Principles
n3 ,
To determine
determine 
(n+1)4 – n4
Utilize the relation obtained for
 n2,  n and  1

28. Addition of Sequences of Numbers: First Principles
(n+1)4 – n4 =  N
n=1
(N+1)4 - 14
The last: n = N+1 &
The first n = 1
The surviving terms:
All other terms cancel out

29. Addition of Sequences of Numbers: First Principles
(n+1)4 – n4 =  N
n=1
(N+1)4 - 14
= N4 + 4N3 + 6N2 + 4N + 1 – 1
= N4 + 4N3 + 6N2 + 4N

30. Addition of Sequences of Numbers: First Principles
(n+1)4 – n4 =  N
n=1
N4 + 4N3 + 6N2 + 4N
n4 + 4n3 + 6n2 + 4n + 1- n4, or
n4 + 4n3 + 6n2 + 4n + 1- n4

31. Addition of Sequences of Numbers: First Principles
(n+1)4 – n4 =  N
n=1
N4 + 4N3 + 6N2 + 4N
4n3 + 6n2 + 4n + 1

32. Addition of Sequences of Numbers: First Principles
(n+1)4 – n4  N
n=1
= N4 + 4N3 + 6N2 + 4N
4n3 + 6n2 + 4n + 1

33. Addition of Sequences of Numbers: First Principles
4n3 + 6n2 + 4n + 1
= N4 + 4N3 + 6N2 + 4N

34. Addition of Sequences of Numbers: First Principles
4n3 + 6n2 + 4n + 1
= N4 + 4N3 + 6N2 + 4N
6N3 6N2 6N
4N(N+1) 2 N

35. Addition of Sequences of Numbers: First Principles
4n3 + 6n2 + 4n + 1
= N4 + 4N3 + 6N2 + 4N
6N3 6N2 6N
4N2 + 4N 2 N

36. Addition of Sequences of Numbers: First Principles
4n3 + 6n2 + 4n + 1
= N4 + 4N3 + 6N2 + 4N
6N3 6N2 6N
4N2 + 4N 2 N

37. Addition of Sequences of Numbers: First Principles
4n3 + 6n2 + 4n + 1
= N4 + 4N3 + 6N2 + 4N
6N3 5N2 6N 4N 2 N

38. Addition of Sequences of Numbers: First Principles
4n3 + 6n2 + 4n + 1
= N4 + 4N3 + 6N2 + 4N
6N3 5N2 4N 3

39. Addition of Sequences of Numbers: First Principles
4n3 + 6n2 + 4n + 1
= N4 + 4N3 + 6N2 + 4N
2N3 + 5N2 + 4N

40. Addition of Sequences of Numbers: First Principles
4n3
+ 2N3 + 5N2 + 4N =
N4 + 4N3 + 6N2 + 4N

41. Addition of Sequences of Numbers: First Principles
4n3
+ 2N3 + 5N2 + 4N =
N4 + 4N3 + 6N2 + 4N

42. Addition of Sequences of Numbers: First Principles
4n3
+ 2N3 + 5N2 =
N4 + 4N3 + 6N2

43. Addition of Sequences of Numbers: First Principles
4n3
+ 2N3 + 5N2 =
N4 + 4N3 + 6N2

44. Addition of Sequences of Numbers: First Principles
4n3
+ 2N3 + 5N2 =
N4 + 4N3 + 6N2

45. Addition of Sequences of Numbers: First Principles
4n3
+ 2N3 + 5N2 =
N4 + 2N3 + 6N2

46. Addition of Sequences of Numbers: First Principles
4n3 =
N4 + 2N3 + N2

47. Addition of Sequences of Numbers: First Principles
4n3 =
N4 + 2N3 + N2 4
[N(N+1)]2 4 =

48. Addition of Sequences of Numbers: First Principles
Advantage
Derivation: Does not require any prior
knowledge on the compact form.
nk relies on the knowledge of binomial
expansion, and the compact relationship
derived for nk-1.