Problem 8 - 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. \(S_{n}: 2+7+12+\cdots+(5 n-3)=\frac{n(5 n-1)}{2}\)

Step-by-Step Solution

Answer

The statement \(S_{k}\) is \(2+7+12+\cdots+(5 k-3)=\frac{k(5 k-1)}{2}\) and the statement \(S_{k+1}\) is \(2+7+12+\cdots+(5 k-3)+(5(k+1)-3)=\frac{(k+1)(5(k+1)-1)}{2}\).

Step 1: Define Statement \(S_{k}\)

We start by substituting \(n=k\) into \(S_{n}\), which gives us the equation for \(S_{k}: 2+7+12+\cdots+(5 k-3)=\frac{k(5 k-1)}{2}\)

Step 2: Define Statement \(S_{k+1}\)

Next, we construct \(S_{k+1}\). To do this, we add the next term in the series to \(S_{k}\), which is \(5(k+1)-3\). Thus, \(S_{k+1}: 2+7+12+\cdots+(5 k-3)+(5(k+1)-3)=\frac{k(5 k-1)}{2} + 5(k+1)-3\)

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

To simplify \(S_{k+1}\), we first simplify the right side of the equation \(= \frac{k(5 k-1)}{2} + 5(k+1)-3 = \frac{5 k^{2} - k}{2} + 5k +5 -3 = \frac{5 k^{2} + 9k + 2}{2}\). Then we observe that this is the same as \(\frac{(k+1)(5(k+1)-1)}{2}\) which corresponds to \(S_{n}\) where \(n=(k+1)\), further proving the validity of the original \(S_{n}\) equation by induction.

Key Concepts

Sequences and Series

In mathematics, the terms "sequences" and "series" are related but distinct concepts. A **sequence** is an ordered list of numbers following a specific pattern, while a **series** is the sum of the terms of a sequence.
Sequences can be finite or infinite, and often we denote the terms of a sequence as something like \(a_1, a_2, a_3, \ldots\). In our original exercise, the sequence is composed of the series expression \(2, 7, 12, \ldots, (5n-3)\), which follows a specific linear pattern.

• **Arithmetic sequences** have a common difference between consecutive terms. The given sequence follows this arithmetic pattern: each term increases by 5.
• **Series** are particularly interesting because they sum these sequences. In arithmetic series, the sum involves not just the pattern of the sequence, but also a formula to simplify computation usually involving the count of terms \(n\).

Understanding sequences and series primarily involves recognizing the pattern and applying the proper summation formula to find expressions like \(S_n\). This allows mathematicians to predict the sum of terms even before computing each one.

Algebraic Expressions

An **algebraic expression** is a mathematical phrase encompassing numbers, variables, and operations. It can represent quantities in formulas and simplify expressions. In our problem, algebraic expressions arise in both the sequence terms and their sum.
The term \(5n - 3\) describes each element in the sequence and holds substantial information:

• **Coefficient**: The number 5 indicates the increment pattern in the sequence.
• **Variable**: \(n\) represents the position of the term in the sequence, allowing the expression to generalize any term.
• **Constant**: The subtraction of 3 tailors the sequence to start at 2.These expressions can be combined and manipulated using algebraic rules to derive sums or simplify into series expressions such as \(\frac{n(5n-1)}{2}\). Simplifying requires combining like terms, working with fractions, or factoring, as demonstrated in the original solution.
  Algebraic expressions form the foundation for solving equations and verifying patterns in sequences and series.

Inductive Reasoning

At the heart of the original exercise lies **inductive reasoning**, a logical process used in mathematics to prove statements about all natural numbers. Inductive reasoning involves two main steps: **the base case** and **the inductive step**.

• **Base Case**: Demonstrates the statement is true for the initial value. For example, proving \(S_1 = 2 = \frac{1(5 \cdot 1 - 1)}{2}\) establishes the foundation.
• **Inductive Step**: Assuming the statement is true for \(n = k\), then showing it holds for \(n = k+1\). This ensures if it is true for one number, it must follow for the next.

In our solution, we first define \(S_k\) and then construct and simplify \(S_{k+1}\) by adding the next sequence term. The simplification shows the expression still aligns with the original formula. Completing these steps verifies our initial formula \(S_n\) holds for any number in the sequence.
Using induction is a powerful method because it verifies infinite possibilities with finite steps, crucial for proofs involving sequences and series.