Addition of Sequences of Numbers: First Principles = 1 + 2 + 3 + 4 +…+ (N-1) + N = ? = 12 + 22 + 32 +42 +…+ N2 = ? n2 n = 1 N = 13 + 23 + 33 +43 +…+ N3 = ? n3 n = 1 N Derivation based on: R. N. Zare, Angular Momentum, Chap. 1, (1988)
Addition of Sequences of Numbers: First Principles (n+1) – n = n = 1 N 2-1 + 3-2 + 4-3 + … + N +(N-1) + N+1-N
Addition of Sequences of Numbers: First Principles (n+1) – n = n = 1 N 2-1 + 3-2 + 4-3 + … + N +(N-1) + N+1-N
Addition of Sequences of Numbers: First Principles (n+1) – n = n = 1 N 2-1 + 3-2 + 4-3 + … + N +(N-1) + N+1-N
Addition of Sequences of Numbers: First Principles This will disappear with the next term (n+1) – n = n = 1 N 2-1 + 3-2 + 4-3 + … + N +(N-1) + N+1-N
Addition of Sequences of Numbers: First Principles This will disappear with the next term (n+1) – n = n = 1 N 2-1 + 3-2 + 4-3 + … + N +(N-1) + N+1-N This disappears with the previous term
Addition of Sequences of Numbers: First Principles (n+1) – n = n = 1 N 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
Addition of Sequences of Numbers: First Principles (n+1) – n = n = 1 N 2-1 + 3-2 + 4-3 + … + N +(N-1) + N+1-N The surviving terms
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+1+1+…+1+1 N1, or
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+1+1+…+1+1 N1, or N
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
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)
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)
Addition of Sequences of Numbers: First Principles N n=1 N(N+2) 2n + N n=1 N n=1 1 N
Addition of Sequences of Numbers: First Principles N n=1 N(N+2) N n=1 2n + N = N2 + 2N
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
Addition of Sequences of Numbers: First Principles n2 , To determine first determine (n+1)3 – n3
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
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
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
Addition of Sequences of Numbers: First Principles (3n2 + 3n + 1) = N n=1 N3 + 3N2 + 3N 3N(N+1) 2 N 3N(N+1)
Addition of Sequences of Numbers: First Principles N n=1 3N(N+1) 2 + N = N3 + 3N(N+1)
Addition of Sequences of Numbers: First Principles N n=1 N = N3 + 3N(N+1) 2 = N3 + 3N2 + 3N 2
Addition of Sequences of Numbers: First Principles N n=1 N = N3 + 3N2 + 3N 2
Addition of Sequences of Numbers: First Principles N n=1 = N3 + 3N2 + N 2 n2 N n=1 = N3 + N2 + N 2 6 3
Addition of Sequences of Numbers: First Principles N n=1 = N3 + N2 + N 2 6 3 = N(N+1)(2N+1) 6
Addition of Sequences of Numbers: First Principles n3 , To determine determine (n+1)4 – n4 Utilize the relation obtained for n2, n and 1
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
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
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
Addition of Sequences of Numbers: First Principles (n+1)4 – n4 = N n=1 N4 + 4N3 + 6N2 + 4N 4n3 + 6n2 + 4n + 1
Addition of Sequences of Numbers: First Principles (n+1)4 – n4 N n=1 = N4 + 4N3 + 6N2 + 4N 4n3 + 6n2 + 4n + 1
Addition of Sequences of Numbers: First Principles 4n3 + 6n2 + 4n + 1 = N4 + 4N3 + 6N2 + 4N
Addition of Sequences of Numbers: First Principles 4n3 + 6n2 + 4n + 1 = N4 + 4N3 + 6N2 + 4N 6N3 6N2 6N 3 2 6 + + 4N(N+1) 2 N
Addition of Sequences of Numbers: First Principles 4n3 + 6n2 + 4n + 1 = N4 + 4N3 + 6N2 + 4N 6N3 6N2 6N 3 2 6 + + 4N2 + 4N 2 N
Addition of Sequences of Numbers: First Principles 4n3 + 6n2 + 4n + 1 = N4 + 4N3 + 6N2 + 4N 6N3 6N2 6N 3 2 6 + + 4N2 + 4N 2 N
Addition of Sequences of Numbers: First Principles 4n3 + 6n2 + 4n + 1 = N4 + 4N3 + 6N2 + 4N 6N3 5N2 6N 3 6 + + 4N 2 N
Addition of Sequences of Numbers: First Principles 4n3 + 6n2 + 4n + 1 = N4 + 4N3 + 6N2 + 4N 6N3 5N2 4N 3 + +
Addition of Sequences of Numbers: First Principles 4n3 + 6n2 + 4n + 1 = N4 + 4N3 + 6N2 + 4N 2N3 + 5N2 + 4N
Addition of Sequences of Numbers: First Principles 4n3 + 2N3 + 5N2 + 4N = N4 + 4N3 + 6N2 + 4N
Addition of Sequences of Numbers: First Principles 4n3 + 2N3 + 5N2 + 4N = N4 + 4N3 + 6N2 + 4N
Addition of Sequences of Numbers: First Principles 4n3 + 2N3 + 5N2 = N4 + 4N3 + 6N2
Addition of Sequences of Numbers: First Principles 4n3 + 2N3 + 5N2 = N4 + 4N3 + 6N2
Addition of Sequences of Numbers: First Principles 4n3 + 2N3 + 5N2 = N4 + 4N3 + 6N2
Addition of Sequences of Numbers: First Principles 4n3 + 2N3 + 5N2 = N4 + 2N3 + 6N2
Addition of Sequences of Numbers: First Principles 4n3 = N4 + 2N3 + N2
Addition of Sequences of Numbers: First Principles 4n3 = N4 + 2N3 + N2 4 [N(N+1)]2 4 =
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.