Solving the Summation: From i0 to n-2, n - 1 - i
In this article, we explore a specific summation problem often encountered in mathematical contexts. Specifically, we aim to solve the sum from (i0) to (n-2)
∑i0to(n-2)(n-1-i)
Reformulating the Problem in LaTeX
To make the problem more clear, let's reformulate it using LaTeX:
∑i0to(n-2)(n-1-i)Mathematically, this can be rewritten as:
S∑i0to(n-2)(n-1-i)By shifting the index (i), we can simplify the expression as follows:
S∑i1to(n-1)(n-i)
Breaking Down the Summation
The problem can be broken down into two parts, each significantly simpler to solve:
The summation of (n) over the range from 1 to (n-1) The summation of (i) over the range from 1 to (n-1)Let's evaluate each part separately:
Part 1: Summation of (n)
The first part is:
N∑n1to(n-1)nThis simplifies to:
Nn×(n-1)Part 2: Summation of (i)
The second part is:
I∑n1to(n-1)∑iThe sum of the first (n-1) integers can be found using the formula:
∑1to(n-1)imm-12×(n-1)
Where (m n-1):
In-12×(n-1)
Final Calculation
By combining both parts, we get:
SN-In×(n-1)-n-12×(n-1)Which simplifies to:
S2nmi-3-12×(n-1)n-12×(2n-n-1)Which further simplifies to:
Sn-12×(n-1)n(n-1)2-(n-1)(n-1)2n(n-1)-(n-1)(n-1)2n-12Therefore, the result of the summation is:
Sn-12
Conclusion
This solution shows us that the summation of (n - 1 - i) from (i0) to (n-2) can be simplified significantly, and the result is intriguingly symmetrical. By understanding the breakdown and simplification, we can tackle similar problems more effectively.
References
[1] Wikipedia - Summation
[2] MathWorld - Sum