Problem 6 - 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. \(S_{n}: 3+4+5+\cdots+(n+2)=\frac{n(n+5)}{2}\)

Step-by-Step Solution

Answer

The statements are \(S_{k}: 3+4+5+\cdots+(k+2)=\frac{k(k+5)}{2}\) and \(S_{k+1}: 3+4+5+\cdots+(k+3)=\frac{(k+1)(k+6)}{2}\)

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

To create \(S_{k}\), replace all instances of \(n\) in \(S_{n}\) with \(k\). This yields the equation: \(3+4+5+\cdots+(k+2)=\frac{k(k+5)}{2}\)

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

To derive \(S_{k+1}\), replace \(n\) with \(k+1\) in \(S_{n}\). This will result in equation: \(3+4+5+\cdots+[(k+1)+2]=\frac{(k+1)(k+6)}{2}\)

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

By simplifying the right side of the equation, we get: \(3+4+5+\cdots+(k+3)=\frac{(k+1)(k+6)}{2}\), this is the completely simplified form of \(S_{k+1}\)

Key Concepts

Sequence Summation

Sequence summation is a technique used to find the total or sum of elements within a sequence. In mathematics, especially when dealing with arithmetic sequences, summation is a common task. Here, we focus on understanding sequences like \(3, 4, 5, \ldots, (n+2)\). The sequence starts at 3 and ends at \(n+2\), with each term incrementing by 1.
To find the sum of such sequences, one may use formulas tailored for arithmetic series. The formula for the sum of an arithmetic sequence is:

• Sum \( = \frac{ ext{number of terms}}{2} \times ( ext{first term} + ext{last term}) \)

In the given exercise, this sum formula transforms into a form that helps prove sequences using mathematical induction. By replacing \(n\) with \(k\) and \(k+1\), one can derive simplifications necessary for proving properties about sums. Understanding sequence summation is essential in algebra to solve many types of numerical problems, especially in academic contexts.

Algebraic Expressions

Algebraic expressions form the foundation for translating word problems or sequences into mathematical language, which one can then manipulate. In the exercise, expressions like \(\frac{n(n+5)}{2}\) or \(\frac{(k+1)(k+6)}{2}\) serve as algebraic representations of summed sequences.
Understanding these expressions involves recognizing that they consist of numbers, variables (like \(n\) or \(k\)), and operational symbols. These expressions allow students to rewrite sequences in simpler forms. They are crucial for applying further algebraic manipulation, such as simplification or solving for particular variables.
The primary goal is to be able to derive one expression from another, ensuring the same value, or proving they represent the same quantity in different circumstances. Practice in simplifying and rearranging algebraic expressions strengthens problem-solving abilities and affords students versatility in handling diverse mathematical challenges.

Integer Sequences

Integer sequences are lists of numbers following a specific pattern or arithmetic rule. In the context of this exercise, the sequence outlined by \(3, 4, 5, \ldots, (n+2)\) is a straightforward example where each successive integer increases by 1.
Such sequences are important in mathematics because they allow patterns to emerge, which can make complex calculations more predictable and manageable. Integer sequences are often used in proofs, especially induction, since they showcase how formulas behave when components of the sequence expand or continue indefinitely.

• Arithmetic sequences: These have a constant difference between successive terms, as demonstrated in the given sequence.
• Geometric sequences: These have a constant ratio between successive terms.

In the given problem, recognizing the pattern of increasing integers from \(3\) to \(n+2\) helps in deploying methods like induction to prove formula correctness. With integer sequences, comprehending the progression aids in planning and executing proofs, ultimately contributing to a broader understanding of algebraic structures.