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Addition of Sequences of Numbers: First Principles

= 1 + 2 + 3 + 4 +…+ (N-1) + N = ?. = 1 2 + 2 2 + 3 2 +4 2 +…+ N 2 = ?. N. N. N.  n.  n 2.  n 3. n = 1. n = 1. n = 1. = 1 3 + 2 3 + 3 3 +4 3 +…+ N 3 = ?. Addition of Sequences of Numbers: First Principles.

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Addition of Sequences of Numbers: First Principles

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  1. = 1+ 2+ 3+ 4+…+ (N-1) + N= ? = 12 + 22 + 32 +42 +…+ N2 = ? N N N  n  n2  n3 n = 1 n = 1 n = 1 = 13 + 23 + 33 +43 +…+ N3 = ? Addition of Sequences of Numbers: First Principles Derivation based on: R. N. Zare, Angular Momentum, Chap. 1, (1988)

  2. N • (n+1) – n = n = 1 Addition of Sequences of Numbers: First Principles 2-1 + 3-2 + 4-3 + … + N +(N-1) + N+1-N

  3. N • (n+1) – n = n = 1 Addition of Sequences of Numbers: First Principles 2-1 + 3-2 + 4-3 + … + N +(N-1) + N+1-N

  4. N • (n+1) – n = n = 1 Addition of Sequences of Numbers: First Principles 2-1 + 3-2 + 4-3 + … + N +(N-1) + N+1-N

  5. N • (n+1) – n = n = 1 This will disappear with the next term Addition of Sequences of Numbers: First Principles 2-1 + 3-2 + 4-3 + … + N+(N-1) + N+1-N

  6. N • (n+1) – n = n = 1 This will disappear with the next term This disappears with the previous term Addition of Sequences of Numbers: First Principles 2-1 + 3-2 + 4-3 + … + N+(N-1) + N+1-N

  7. N • (n+1) – n = n = 1 Addition of Sequences of Numbers: First Principles 2-1 + 3-2 + 4-3 + … + N+(N-1) + N+1-N This will disappear with the next term This disappears with the previous term

  8. N • (n+1) – n = n = 1 Addition of Sequences of Numbers: First Principles 2-1 + 3-2 + 4-3 + … + N+(N-1) + N+1-N The surviving terms

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

  34. 6N36N26N 3 2 6 + + 4N(N+1) 2 Addition of Sequences of Numbers: First Principles N n=1  4n3+6n2+4n+1 = N4+4N3+6N2+4N N

  35. 6N36N26N 3 2 6 + + 4N2 + 4N 2 Addition of Sequences of Numbers: First Principles N n=1  4n3+6n2+4n+1 = N4+4N3+6N2+4N N

  36. 6N36N26N 3 2 6 + + 4N2 + 4N 2 Addition of Sequences of Numbers: First Principles N n=1  4n3+6n2+4n+1 = N4+4N3+6N2+4N N

  37. Addition of Sequences of Numbers: First Principles N n=1  4n3+6n2+4n+1 = N4+4N3+6N2+4N 6N35N26N 3 6 + + 4N 2 N

  38. Addition of Sequences of Numbers: First Principles N n=1  4n3+6n2+4n+1 = N4+4N3+6N2+4N 6N35N24N 3 + +

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

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

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

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

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

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

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

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

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

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

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