Problem 1 - Algebra and Trigonometry

Question

Express the given quantity in terms of the indicated variable. The sum of three consecutive integers; \(\quad n=\) first integer of the three

Step-by-Step Solution

Answer

The sum is \( 3n + 3 \).

Step 1: Define the consecutive integers

To express the sum of three consecutive integers where \( n \) is the first integer, we need to understand that if \( n \) is the first integer, the next two integers will be \( n+1 \) and \( n+2 \).

Step 2: Determine the expression for the sum

Add the three consecutive integers together to find the expression for their sum. This gives us:\[ n + (n+1) + (n+2) \]

Step 3: Simplify the expression

Combine like terms in the expression:\[ n + (n+1) + (n+2) = n + n + 1 + n + 2 = 3n + 3 \]

Step 4: Finalize the expression

The sum of the three consecutive integers in terms of \( n \) is:\[ 3n + 3 \].

Key Concepts

Algebraic Expressions

Algebraic expressions are like a mathematical shorthand, allowing us to simplify and solve problems efficiently. In our problem, the given expression of the sum of three consecutive integers is represented using the variable \( n \). This is a typical use of algebraic expressions—taking a situation described in words and translating it into numbers and symbols.
For example, the first integer is represented by \( n \), making the next two consecutive integers \( n+1 \) and \( n+2 \). Each part of this equation is an algebraic expression, expressing the idea that each integer directly follows the last.

• Algebraic expressions are versatile tools for showing relationships between numbers.
• They use variables (like \( n \) in this case) to hold values that can change within the context.
• This helps us generalize problems, like finding the sum of any three consecutive integers starting at \( n \).

By representing the sum with \( n + (n+1) + (n+2) \), we're able to simplify it down to the more compact formula \( 3n + 3 \), ensuring it is easy to use and understand in further calculations.

Integer Sequence

Integer sequences follow a specific order or pattern, making them a great way to learn about numerical patterns and relationships. In this exercise, the idea of consecutive integers highlights the concept of a sequence in which each number is one more than the previous.
Specifically, if the first integer in the sequence is \( n \), then the entire sequence is \( n, n+1, n+2 \). This sequence clearly shows the defining characteristic of consecutive integers.

• Integer sequences are simple but powerful; they demonstrate incremental progression.
• In consecutive sequences, each integer can be derived from a formula involving the previous one.

Understanding integer sequences can help you recognize underlying patterns in larger sets of numbers, making it easier to predict future numbers in the sequence or to find sums, like in our problem.

Sum of Integers

The sum of integers is a fundamental part of mathematics, and consecutive integers offer a straightforward introduction to this concept. Adding consecutive integers operates under a clear, predictable pattern because each integer simply increases by one over the previous.
To compute the sum here, we take the formula \( n + (n+1) + (n+2) \), which translates into the simplified algebraic expression \( 3n + 3 \). This pattern is reliable enough to extend to sums involving any number of consecutive integers.

• Summing integers typically involves adding a series of similar terms.
• When terms are consecutive, patterns like adding 1 to each term help provide quick solutions.
• The simplicity of the pattern makes it easy to find sums or construct formulas like \( 3n + 3 \).

Learning how to combine consecutive integers efficiently with algebraic expressions forms a solid basis for tackling more complex numerical problems.