Addition of Sequences of Numbers: First Principles
= 12 + 22 + 32 +42 +…+ N2 = ? n2
n = 1
N
= 13 + 23 + 33 +43 +…+ N3 = ? n3
n = 1
N
nn = 1
N
= 1 + 2 + 3 + 4 +…+ (N-1) + N = ?
Derivation based on: R. N. Zare, Angular Momentum, Chap. 1, (1988)
(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
(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
This disappears with the previous term
(n+1) – n =n = 1
N
Addition of Sequences of Numbers: First Principles
2-1 + 3-2 + 4-3 + … + N +(N-1) + N+1-N
This disappears with the previous term
This will disappear with the next term
The surviving terms
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
N+1-1
= N
(n+1) – n =n = 1
N
N1, or
n = 1
N
1 = 1+1+1+1+…+1+1
Addition of Sequences of Numbers: First Principles
(n+1) – n =n = 1
N
N+1-1
= N
(n+1) – n =n = 1
N
N1, or
n = 1
N
1 = 1+1+1+1+…+1+1
N
Addition of Sequences of Numbers: First Principles
The surviving terms:
(n+1)2 – n2 =N
n=1
(N+1)2 - 12
All other terms cancel out
The last: n = N+1 & The first n = 1
Addition of Sequences of Numbers: First Principles
(n+1)2 – n2 =N
n=1
(N+1)2 - 12
= N2 + 2N + 1 – 1
= N2 + 2N = N(N+2)
= N2 + 2N + 1 – 1
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
(2n + 1) =N
n=1
N(N+2)
N
n=1
1
N
2n + N
n=1
Addition of Sequences of Numbers: First Principles
(2n + 1) =N
n=1
N(N+2)
2n + N
n=1
N = N2 + 2N
Addition of Sequences of Numbers: First Principles
(2n + 1) =N
n=1
N(N+2)
2nN
n=1
= N2 + N = N(N+1)
nN
n=1
= N(N+1)2
Addition of Sequences of Numbers: First Principles
To determine n2 , first determine
(n+1)3 – n3
Addition of Sequences of Numbers: First Principles
The surviving terms:
(n+1)3 – n3 =N
n=1
(N+1)3 - 13
All other terms cancel out
The last: n = N+1 & The first n = 1
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 + 1 – n3
n3 + 3n2 + 3n + 1 – n3
(3n2 + 3n + 1) =N
n=1
N3 + 3N2 + 3N
N3 + 3N2 + 3N
Addition of Sequences of Numbers: First Principles
(3n2 + 3n + 1) =N
n=1
N3 + 3N2 + 3N
N3N(N+1)
2
3N(N+1)
Addition of Sequences of Numbers: First Principles
(3n2) +N
n=1
+ N3N(N+1)
2
= N3 + 3N(N+1)
Addition of Sequences of Numbers: First Principles
(3n2) +N
n=1
N 3N(N+1)
2
= N3 +
= N3 + 3N2 + 3N2 2
Addition of Sequences of Numbers: First Principles
(3n2) +N
n=1
N = N3 + 3N2 + 3N2 2
Addition of Sequences of Numbers: First Principles
(3n2)N
n=1
= N3 + 3N2 + N2 2
n2N
n=1
= N3 + N2 + N2 63
Addition of Sequences of Numbers: First Principles
n2N
n=1
= N3 + N2 + N2 63
= N(N+1)(2N+1) 6
Addition of Sequences of Numbers: First Principles
To determine n3 , determine
(n+1)4 – n4
Utilize the relation obtained for
n2, n and 1
Addition of Sequences of Numbers: First Principles
The surviving terms:
(n+1)4 – n4 =N
n=1
(N+1)4 - 14
All other terms cancel out
The last: n = N+1 & The first n = 1
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 – n4N
n=1
= N4 + 4N3
+ 6N2 + 4N
4n3 + 6n2 + 4n + 1
Addition of Sequences of Numbers: First Principles
N
n=1
= N4 + 4N3
+ 6N2 + 4N4n3 + 6n2 + 4n + 1
Addition of Sequences of Numbers: First Principles
N
n=1
4n3 + 6n2 + 4n + 1 = N4 + 4N3
+ 6N2 + 4N
N4N(N+1)
2
6N3 6N2 6N 3 2 6
+ +
Addition of Sequences of Numbers: First Principles
N
n=1
4n3 + 6n2 + 4n + 1 = N4 + 4N3
+ 6N2 + 4N
N4N2 + 4N
2
6N3 6N2 6N 3 2 6
+ +
Addition of Sequences of Numbers: First Principles
N
n=1
4n3 + 6n2 + 4n + 1 = N4 + 4N3
+ 6N2 + 4N
N4N2 + 4N
2
6N3 6N2 6N 3 2 6
+ +
Addition of Sequences of Numbers: First Principles
N
n=1
4n3 + 6n2 + 4n + 1 = N4 + 4N3
+ 6N2 + 4N
N 4N2
6N3 5N2 6N 3 6
+ +
Addition of Sequences of Numbers: First Principles
N
n=1
4n3 + 6n2 + 4n + 1 = N4 + 4N3
+ 6N2 + 4N
6N3 5N2 4N 3
+ +
Addition of Sequences of Numbers: First Principles
N
n=1
4n3 + 6n2 + 4n + 1 = N4 + 4N3
+ 6N2 + 4N
2N3 + 5N2 + 4N
Addition of Sequences of Numbers: First Principles
N
n=1
4n3
N4 + 4N3
+ 6N2 + 4N
+ 2N3 + 5N2 + 4N =
Addition of Sequences of Numbers: First Principles
N
n=1
4n3
N4 + 4N3
+ 6N2 + 4N
+ 2N3 + 5N2 + 4N =
Addition of Sequences of Numbers: First Principles
N
n=1
4n3 N4 + 4N3
+ 6N2+ 2N3 + 5N2 =
Addition of Sequences of Numbers: First Principles
N
n=1
4n3 N4 + 4N3
+ 6N2+ 2N3 + 5N2 =
Addition of Sequences of Numbers: First Principles
N
n=1
4n3 N4 + 4N3
+ 6N2+ 2N3 + 5N2 =
Addition of Sequences of Numbers: First Principles
N
n=1
4n3 N4 + 2N3
+ 6N2+ 2N3 + 5N2 =
Addition of Sequences of Numbers: First Principles
N
n=1
4n3 N4 + 2N3
+ N2=
Addition of Sequences of Numbers: First Principles
N
n=1
4n3 N4 + 2N3
+ N2=
4
[N(N+1)]2
4=
Addition of Sequences of Numbers: First Principles
Derivation: Does not require any priorknowledge on the compact form.
nk relies on the knowledge of binomialexpansion, and the compact relationshipderived for nk-1.
Advantage
