Problem 5 - Algebra and Trigonometry

Question

A statement \(S_{n}\) about the positive integers is given. Write statements \(S_{k}\) and \(S_{k+1},\) simplifying statement \(S_{k+1}\) completely. \(S_{n}: 4+8+12+\cdots+4 n=2 n(n+1)\)

Step-by-Step Solution

Answer

The statements are \(S_{k} : 4 + 8 + 12 + \cdots + 4k = 2k(k + 1)\) and \(S_{k+1} : 4 + 8 + 12 + \cdots + 4k + 4(k + 1) = 2(k+1)(k+2)\)

Step 1: Formulate the Statement \(S_{k}\)

Statement \(S_{k}\) for the given series would be expressed as: \(S_{k} : 4 + 8 + 12 + \cdots + 4k = 2k(k + 1)\)

Step 2: Formulate the Statement \(S_{k+1}\) before Simplification

We express the next term of the series as \(4(k+1)\) and it is added to both sides of equation \(S_{k}\) to achieve \(S_{k+1}\): \(S_{k+1} : 4 + 8 + 12 + \cdots + 4k + 4(k + 1) = 2k(k + 1) + 4(k+1)\)

Step 3: Simplify the Statement \(S_{k+1}\)

The right-hand side of the equation is simplified, which results in: \(S_{k+1} : 4 + 8 + 12 + \cdots + 4k + 4(k + 1) = 2(k+1)(k+2)\)

Key Concepts

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and mathematical operations, forming the building blocks of algebra. These expressions are crucial for representing real-world situations in a mathematical form. Let's take a closer look at the expression involved in our exercise:

• The expression in question is a series of terms like \(4n\), where \(n\) is a variable representing the position of each term in the sequence.
• In the context of our exercise, we're given a series starting from 4 and incremented by 4 in each succeeding term, creating the algebraic expression \(4 + 8 + 12 + \cdots + 4n\).

At first glance, the given series is just a simple sum of multiples of four. However, it beautifully simplifies into a quadratic expression, represented as \(2n(n+1)\). Understanding this transformation is key to mastering algebraic expression manipulations. Each term in the sequence, when summed properly, can be rewritten in a more generalized form that makes further algebraic operations easier.
This restructuring is vital when working with sequences, allowing us to explore properties of the series easily and efficiently.

Series and Sequences

Series and sequences are fundamental components of mathematics, particularly when discussing infinite, finite, or converging collections of numbers. In our exercise, we're dealing with an arithmetic series — a sequence of numbers with a common difference between consecutive terms. Here's how it applies:

• The given series \(4 + 8 + 12 + \cdots + 4n\) is arithmetic because each term increases by a constant value of 4.
• The series can be generalized and expressed in summation notation, reflecting its structured progression.

When transformed into the equation \(2n(n+1)\), we get a step closer to understanding the symmetry and beauty behind the progression. Recognizing the patterns in sequences helps in predicting further values and provides a method for finding the sum of large series efficiently.
Also, becoming skilled in working with series enables you to handle more complex mathematical problems involving infinite sequences and their convergent properties, opening doors to advanced calculus and beyond.

Simplification in Algebra

Simplification in algebra involves transforming complex expressions into simpler or more efficient forms without changing their value. In the exercise given, the core task was simplifying the expression on the right of the equation \(S_{k+1}\). Examine the transformation:

• Starting with \(2k(k+1) + 4(k+1)\), simplifying involves identifying common factors and terms.
• Factoring out common elements, we rewrite the expression as \(2(k+1)(k + 2)\).

This not only makes the expression neater but also more comprehensible for further operations like substitution or derivation of further sequences. Simplification is crucial, as it lays the groundwork for solving equations or proving mathematical statements.
By consistently practicing simplification, you become proficient in seeing through complex algebraic expressions, allowing you to tackle difficult problems with ease. Moreover, with simplification, unnecessary computations are reduced, saving time and effort in larger, more extensive calculations.