Question

Prove that $$ 1 \cdot 2 \cdot 3+2 \cdot 3 \cdot 4+\cdots+n \cdot(n+1) \cdot(n+2)=n(n+1)(n+2)(n+3) / 4 $$ This problem should not be solved using a proof by induction.

Short answer

Evaluate and simplify the sequence using known sums to prove the given formula.

Step-by-step solution

Understand the Problem

The problem requires us to find the sum of a sequence which is composed of products of three consecutive integers, starting from 1. We must show that this sum equals \( \frac{n(n+1)(n+2)(n+3)}{4} \).

Evaluate the Sequence

The sequence can be written as a sum: \( \sum_{k=1}^{n} k(k+1)(k+2) \). We need to find a formula for this sum.

Expand a Single Term

Consider the general term \( k(k+1)(k+2) \). Expand this for a better understanding:\[ k(k+1)(k+2) = k(k^2 + 3k + 2) = k^3 + 3k^2 + 2k \]

Use Known Summation Formulas

Utilize known summation formulas:- \( \sum_{k=1}^{n} k = \frac{n(n+1)}{2} \)- \( \sum_{k=1}^{n} k^2 = \frac{n(n+1)(2n+1)}{6} \)- \( \sum_{k=1}^{n} k^3 = \left(\frac{n(n+1)}{2}\right)^2 \)Then find the sum of each component in the expanded term across all terms in the sequence.

Sum Each Component

Using the expanded form from Step 3, calculate the sum:\[ \sum_{k=1}^{n} k^3 = \left(\frac{n(n+1)}{2}\right)^2 \]\[ \sum_{k=1}^{n} 3k^2 = 3 \frac{n(n+1)(2n+1)}{6} \]\[ \sum_{k=1}^{n} 2k = 2 \frac{n(n+1)}{2} \].

Combine Sums

Combine these results to get:\[ \sum_{k=1}^{n} \left(k^3 + 3k^2 + 2k\right) = \left(\frac{n(n+1)}{2}\right)^2 + \frac{3n(n+1)(2n+1)}{6} + n(n+1) \].

Simplify the Combined Expression

Simplify the combined expression using algebra to reach the closed form \( \frac{n(n+1)(n+2)(n+3)}{4} \). This concludes the proof.

Key concepts

Discrete Mathematics

Discrete Mathematics is a branch of mathematics that deals with distinct and separate values, meaning values that are not connected or continuous. In this context, when we talk about summation formulas, we are essentially dealing with methods to find the total or sum of values that have distinct differences between them. These values are often arranged or indexed by numbers in series and sequences.

One primary area of discrete mathematics is understanding how sequences behave and what formulas we can use to represent and manipulate them. The sequence in the original exercise is a perfect example of this, as it explores the sums of the products of consecutive integers.

By using known summation formulas, we can solve complex problems in discrete mathematics without relying solely on manual calculations. This involves, for instance, breaking down the problem into simpler components, such as individual summations of sequences of powers, like sums of cubes, squares, or linear sequences. Applying these known approaches allows for effective and efficient solutions to problems that appear daunting at first glance.

Polynomial Expansion

Polynomial expansion involves breaking down algebraic expressions that involve multiple coefficients and variables raised to powers. The concept is about expanding and simplifying expressions to make them more manageable and solvable.

The original exercise task required the expansion of a product like \(k(k+1)(k+2)\). This expression was first expanded to produce the polynomial \(k^3 + 3k^2 + 2k\). Through polynomial expansion, each term can be examined and analyzed separately, giving a clearer view of the relationships within the sequence.

This technique is exceptionally useful in solving problems involving summation of sequences. By expanding each term, and then applying known summation formulas, we can derive solutions without performing the explicit summation of each sequence value. Thus, polynomial expansion provides an insightful approach toward solving a broad range of mathematical problems efficiently.

Series and Sequences

In mathematics, series and sequences are essential concepts that describe ordered lists of numbers. Each number is called a term, and these terms can follow specific rules or patterns. Understanding these patterns allows us to find the sum of sequences efficiently.

In the problem we have, the sequence is part of a broader category where terms increase based on specific formulae. Particularly, the series \(\sum_{k=1}^{n} k(k+1)(k+2)\) involves calculating the sum of products of sequential integers.

To solve such problems, one often uses various mathematical tools like known summation formulas, which are crucial for breaking down and evaluating each component sum. This might include using the harmonic sum for linear terms, Gaussian sum formulas for squares, or summation formulas for cubes.

Series and sequences are fundamental topics in both secondary and tertiary level math education. They form the building blocks for understanding more complex mathematical principles, such as calculus and advanced algebra.