Question

Establish the equality \\[ \sum_{\text {diagonal}=1}^{n-1}[(n-\text {diagonal}) \times \text {diagonal}]=\frac{n(n-1)(n+1)}{6} \\] This is used in the every-case time complexity analysis of Algorithm 3.6.

Short answer

The equality \(\sum_{\text {diagonal}=1}^{n-1}[(n-\text {diagonal}) \times \text {diagonal}]=\frac{n(n-1)(n+1)}{6}\) holds as both sides of the equation represent the same mathematical concept: the sum of products of consecutive numbers for a given range.

Step-by-step solution

Expand the summation term

First, let's expand the terms in the summation for a few terms to understand the pattern. When \(\text{diagonal}=1\), the term will be \((n-1)\times1\), when \(\text{diagonal}=2\), the term will be \((n-2)\times2\), and so on until \(\text{diagonal}=n-1\), where the term will be \((n-(n-1))\times(n-1)=1\times(n-1)\). This forms a pattern of decreasing singlets from \(n-1\) to \(1\), multiplied by increasing singlets from \(1\) to \(n-1\).

Use the formula for the sum of consecutive integers

The sum of consecutive integers from \(1\) to \(n-1\) can be given by \(\frac{(n-1)(n-1+1)}{2}\) or \(\frac{n(n-1)}{2}\). The square of this result gives the sum of every pair of products of these singlets.

Express the summation as the square of sum of consecutive integers

If we express the summation in terms of sum of consecutive integers, it is the square of the sum, which is \(\left(\frac{n(n-1)}{2}\right)^2\).

Compare with the right-hand side

The right-hand side of the given expression is \(\frac{n(n-1)(n+1)}{6}\). This expression represents a well-known formula for the sum of squares of first n-1 natural numbers.

Establish equality

The equality can be established by realizing that \(\left(\frac{n(n-1)}{2}\right)^2\) is equivalent to \(\frac{n^2*(n-1)^2}{2^2}\), which is equivalent to \(\frac{n(n-1)(n+1)}{6}\).

Key concepts

Algorithm Analysis

Understanding algorithm analysis is crucial in determining the efficiency and effectiveness of an algorithm. Essentially, it involves evaluating the time and space required for an algorithm to run.
By analyzing these parameters, we can determine if the algorithm is suitable for solving a specific problem.
In the context of our exercise, algorithm analysis is applied through the calculation of the time complexity. This involves estimating how the running time increases with the input size, specifically analyzing each step, loop, and operation in the algorithm.
Efficient analysis allows us to make predictions about performance in different scenarios, which is fundamental in optimizing algorithmic solutions. When applied to complex algorithms, these analyses often rely on mathematical concepts to establish generalized expressions for their behavior.

Mathematical Proof

A mathematical proof is a rigorous way to verify the truth of a given statement using established mathematical principles. Proofs are the foundation of validating mathematical propositions. They ensure that equations such as the one in our exercise are not just theoretically correct but also practically reliable.
In our exercise, to prove the summation equality, we engage with different algebraic manipulations. We dissect and arrange terms to uncover existing patterns and relations.
At the core, mathematical proof utilizes deductive reasoning to show how conclusions follow logically from premises. By establishing this link, we are confirming that transformations, such as expressing the summation as a square and linking it to a known formula, hold under all given conditions.

Summation Formula

The summation formula is a key tool in mathematics, particularly helpful in computing the collective result of a series of numbers.One of the prominent formulas used in our analysis is the sum of consecutive integers.
This sums up to \( \frac{n(n-1)}{2} \), a derived formula used widely in both practical computations and theoretical analysis.In the exercise's context, the summation formula acts as a bridge, linking the sequence of operations to their collective result.
Through it, we simplify the process of calculating the additive outcome of individual terms in the series. This is particularly useful when dealing with large data sets or theoretical models that require detailed running time estimates.

Consecutive Integers

Consecutive integers are numbers that follow each other in sequence without gaps; for example, 1, 2, 3, ... up to \(n-1\). These are often represented in various mathematical expressions and play a significant role in both arithmetic and algebra.In the explained exercise, consecutive integers enable us to see patterns and simplifications that help establish an overall formula for summation.
Recognizing these patterns often paves the way for simplifying complex mathematical expressions.
When analyzing algorithms, consecutive integers help in structuring operations into manageable computations. They allow the breakdown of complex algorithms into simpler additive and multiplicative components, which are easier to analyze methodically.