Rules of consequent addition

  • Thread starter vin300
  • Start date
  • #1
601
4
I have recently discovered some mathematical rules by trial and error. Some of these may be already known to people of mathematics.
1. This one is common: Sum of all integers 1+2+3+...=0.5(n^2 + n)
2. Sum of odd integers: 1+3+5+7+9......= n^2
3. Sum of even integers: 2+4+6+8.....= n^2 + n eg: Sum of first two: 2+4= 2^2 +2 =6
4. Sum of gap odd integers(odd integers with one gap) : 1+5+9+13+17....= (n+1)(n+2) eg. 1+5=(1+1)(1+2)=6
5.a. Sum of gap even integers: 2+6+10+14....= 2n^2 eg 2+6+10= 2(3)^2=18
5.b. Sum of gap even integers: 4+8+12+16...= 4(1+2+3+4+...) eg:4+8+12= 4(1+2+3)= 24
 

Answers and Replies

  • #3
phinds
Science Advisor
Insights Author
Gold Member
2019 Award
16,428
6,716
The algebraic derivations of all these is trivial, so yes, they are known.
 
  • #4
Stephen Tashi
Science Advisor
7,405
1,375
I have recently discovered some mathematical rules by trial and error.
If you want to develop such rules in a systematic fashion, try starting with a function like F(n) = n^2 and forming the series give by its differences: (F(1) - F(0)) + ( F(2) - F(1)) + (F(3) - F(2)) + ....(F(n+1) - F(n)). This series sums to F(n+1) - F(0).

For functions that are simple to write down, the "answer" F(n+1) - F(0) looks simple while the series whose terms are T(i) = F(i+1) - F(i) can look complicated.
 
  • #5
601
4
4. Sum of gap odd integers(odd integers with one gap) : 1+5+9+13+17....= (n+1)(n+2)
Sum of odd integers with one gap = n(2n-1)
 
  • #6
22,089
3,286
Try to find a general formula now which includes all those cases:

[tex]a+ (a+b) + (a + 2b) + (a + 3b) + ... + (a + nb)[/tex]

And try to find the sum of squares:

[tex]1 + 4 + 9 + ... + n^2[/tex]

Or if you're up for a challenge, the more general

[tex]1 + 2^k + 3^k + ... + n^k[/tex]
 

Related Threads on Rules of consequent addition

Replies
2
Views
893
  • Last Post
Replies
1
Views
784
  • Last Post
Replies
23
Views
3K
  • Last Post
Replies
7
Views
515
  • Last Post
Replies
2
Views
1K
  • Last Post
Replies
2
Views
2K
  • Last Post
Replies
7
Views
5K
Replies
14
Views
1K
Top