Problem 66 - Algebra 1

Question

Consecutive odd integers are odd integers listed in order such as \(5,7,\) and 9 . Find three consecutive odd integers whose sum is 111 . Are there three consecutive odd integers whose sum is \(1111 ?\) Explain.

Step-by-Step Solution

Answer

The three consecutive odd integers that sum up to 111 are 35, 37 and 39. As for the sum of 1111, there are no three consecutive odd integers that add up to the sum since the equation yields a decimal number.

Step 1: State the problem as an equation

Let's let the first integer be \(x\). Because odd integers are two units apart, the next consecutive odd numbers will be \(x + 2\) and \(x + 4\). For these three consecutive numbers to add to 111, we can write the equation: \(x + (x + 2) + (x + 4) = 111\)

Step 2: Solve the equation

Solving the equation \(x + (x + 2) + (x + 4) = 111\), we can simplify this to \(3x + 6 = 111\). Further simplifying, we subtract 6 from both sides, we get \(3x = 105\). Dividing by 3, we have \(x = 35\). Therefore, the three consecutive odd integers are 35, 37, and 39.

Step 3: Checking if there are three consecutive odd integers totaling 1111

Let's use the same principle to find if there are three consecutive odd integers that add up to 1111 by setting the equation \(x + (x + 2) + (x + 4) = 1111\). Simplifying, we get \(3x + 6 = 1111\), or \(3x = 1105\), and \(x = 368.33\). However, this is not an integer. Hence, there are no three consecutive odd integers that sum up to 1111.

Key Concepts

Algebraic Equations

Algebraic equations are mathematical statements that express the equality of two expressions. They often include variables, constants, and mathematical operations like addition and multiplication. In our example, we use the equation \(x + (x+2) + (x+4) = 111\) to represent the sum of three consecutive odd integers.When solving algebraic equations, the objective is to find the value of the variables that make the equation true. In this case, we simplified the equation down to \(3x + 6 = 111\) by combining like terms. This process of simplification helps us to see the underlying relationship between the variables and coefficients more clearly. To solve for \(x\), we isolate the variable by performing basic arithmetic operations such as subtracting 6 from both sides and then dividing by 3. Solving algebraic equations often requires a systematic approach to gradually reduce the equation to reveal the value of the unknown variable.

Integer Solutions

Integer solutions are values that satisfy an equation where the variables must take integer values only. An integer is a whole number, which can be positive, negative, or zero, but not a fraction or a decimal.In the exercise, we are looking for integer solutions to the problem, meaning the solution must be a precise whole number. When we calculated \(x = 35\), this was an integer solution for the sum of the integers equaling 111. When trying to solve for a sum of 1111, we found \(x = 368.33\). This is not an integer solution, so it does not meet the problem's criteria.Whenever solving equations that require integer solutions, it's essential to verify whether the operations result in whole numbers. If the operations lead to decimal or fractional outcomes, then the solution is not valid under the requirement of integer solutions.

Sequence and Series

Sequences and series deal with ordered lists of numbers and the sum of these lists, respectively. A sequence is a set of numbers arranged in a specific order, while a series is the sum of the numbers within that sequence.When dealing with sequences of consecutive odd integers, like the ones in our exercise, we observe a consistent difference between each number, which in this case is 2. The first integer plus 2 gives the next integer, and so forth. In a series of consecutive odd integers, each term can be generated from the first term by adding multiples of 2.Finding the sum of a sequence involves recognizing the pattern and formulating it into an equation. As seen, by writing \(x, x+2, x+4\), we transform our number sequence into an algebraic expression that can be easily summed. Understanding sequences and series is crucial in solving problems related to ordered sets of numbers, especially in identifying the sequence patterns and deriving their sums.