Problem 7 - Precalculus Mathematics for Calculus

Question

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

Step-by-Step Solution

Answer

The sum is \(3(n+1)\).

Step 1: Understanding Consecutive Integers

Consecutive integers are numbers that follow each other in order. If the first of these integers is denoted by \(n\), then the next consecutive integers can be represented as \(n+1\) and \(n+2\).

Step 2: Express the Sum of the Integers

The sum of the three consecutive integers is the sum of \(n\), \(n+1\), and \(n+2\). Mathematically, this is expressed as: \(n + (n+1) + (n+2)\).

Step 3: Simplify the Expression

Combine like terms in the expression from Step 2: \(n + (n+1) + (n+2) = n + n + 1 + n + 2 = 3n + 3\).

Step 4: Factor the Expression

Notice that \(3n + 3\) can be factored to simplify it further: \(3(n + 1)\). This expresses the sum in terms of \(n\).

Key Concepts

Consecutive Integers

Consecutive integers are numbers that appear in a sequence without any gaps. For instance, the sequence 3, 4, and 5 are consecutive integers. They follow one another in a predictable pattern, increasing by one. When dealing with consecutive integers, it's vital to denote them with algebraic expressions for clarity.

• If the first integer is represented by \( n \), the next integer becomes \( n + 1 \).
• The one following that is \( n + 2 \), and this pattern continues indefinitely.

Expressing integers in this form is crucial in algebra, as it helps in forming and solving equations involving patterns and sequences.

Sum of Integers

Finding the sum of integers, particularly consecutive ones, is a frequent task. To calculate the sum of three consecutive integers starting with \( n \), you simply add the three expressions:

• The first integer: \( n \)
• The second integer: \( n + 1 \)
• The third integer: \( n + 2 \)

By adding these, you get: \[ n + (n + 1) + (n + 2) \]To simplify this, combine like terms:\[ n + n + 1 + n + 2 = 3n + 3 \]
This simplification facilitates understanding and solving algebraic equations efficiently.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and operations. When expressing the sum of consecutive integers, algebraic expressions form the backbone of this process. They not only represent numbers but also relationships between them. Consider the expression we derived: \[ 3n + 3 \].

• Here, \( 3n \) denotes that \( n \) appears three times.
• The constant \( 3 \) arises from summing the individual integer increments (1 + 2).

Factoring helps in simplifying these expressions further. In our example:

• Factoring \( 3n + 3 \) into \( 3(n + 1) \) reveals a deeper structure.
• It shows how the entire sum can be expressed in terms of the first integer plus one, multiplied by three.

This showcases the power and utility of expressing problems in algebraic terms, as it transforms complex problems into manageable forms.