Problem 13 - Algebra and Trigonometry

Question

Write an algebraic formula for the given quantity.. The sum \(S\) of two consecutive integers, the first integer being \(n\)

Step-by-Step Solution

Answer

The formula is \(S = 2n + 1\).

Step 1: Understand Consecutive Integers

Consecutive integers are integers that follow each other in order. If the first integer is denoted as \(n\), then the next consecutive integer is \(n+1\). This is because the difference between consecutive integers is always 1.

Step 2: Define the Sum

The problem asks for the sum \(S\) of two consecutive integers, beginning with \(n\). Therefore, we set up the expression for the sum as \(S = n + (n + 1)\).

Step 3: Simplify the Expression

Simplify the expression by combining like terms. So, \(S = n + (n + 1) = n + n + 1 = 2n + 1\). This is the algebraic formula for the sum of the two consecutive integers.

Key Concepts

Consecutive Integers

Consecutive integers are numbers that come one after another in a sequence, without skipping any number in between. This means if you start with any integer, the next one in line can be found by simply adding 1 to it. For example, if you start at 5, the next consecutive integer is 6. Here are some key points about consecutive integers:

• They maintain a consistent difference, usually 1, between each number.
• If the first integer is represented as \(n\), the next consecutive integer will be \(n + 1\).
• Understanding how they work is crucial for setting up algebraic expressions involving sequences of numbers.

If you were given \(n = 3\), the consecutive integer sequence would be 3, 4, 5, etc. This predictable nature simplifies problems involving lists of integers.

Simplifying Expressions

Simplifying expressions is a fundamental aspect of algebra, allowing you to express mathematical ideas more clearly. The process involves combining like terms and reducing complexity to make solving equations easier.

For instance, consider the expression for the sum of two consecutive integers, \(S = n + (n + 1)\). Here are the steps to simplify it:

• Write out the expression: \(S = n + (n + 1)\).
• Distribute the terms: \(S = n + n + 1\).
• Combine like terms (in this case, the \(n\) terms): \(S = 2n + 1\).

This simplification process gives us a more concise way of expressing the sum of two consecutive integers, which is invaluable in solving algebraic equations and understanding mathematical relationships more profoundly.

Integer Sum

When calculating the sum of integers, especially consecutive ones, understanding how they add up can simplify your computations. Integer sums appear frequently in mathematics, often forming the basis for more complex problems.

In our context, calculating the sum \(S\) of two consecutive integers starting from \(n\), requires basic operations:

• The first integer is \(n\).
• The second integer, being consecutive, is \(n + 1\).
• Thus, the sum is \(S = n + (n + 1) = 2n + 1\).

This formula \(S = 2n + 1\) quickly gives the sum without listing all steps, proving especially useful when dealing with large integers or sequences.

Algebra

Algebra is a crucial branch of mathematics that deals with symbols and the rules for manipulating those symbols. It allows us to express mathematical relationships and solve various problems by using equations and formulas.

In the example we have, the use of algebra is evident in formulating the expression for the sum of consecutive integers. Here's how algebraic thinking applies:

• We use a variable \(n\) to represent an unknown or general integer.
• The next consecutive integer is expressed as \(n + 1\) using algebraic manipulation.
• The sum \(S\) is formulated algebraically as \(S = 2n + 1\), showing how variables and operations are used to derive solutions.

Mastering algebra involves not just understanding how to set up these expressions but also interpreting and manipulating them to solve broader mathematical problems.