Problem 27 - Prealgebra

Question

Write an equation that describes each sequence. Then find the indicated term. \(7,9,11,13, \dots ; 33\) rd term

Step-by-Step Solution

Answer

The 33rd term of the sequence is 71.

Step 1: Identify the Pattern

Observe the given sequence: 7, 9, 11, 13, ... Notice that the sequence is increasing by a constant difference. The difference between each subsequent term is 2. This tells us the sequence is an arithmetic sequence.

Step 2: Write the General Form of the Arithmetic Sequence

For an arithmetic sequence, the nth term is given by the formula \(a_n = a_1 + (n-1) imes d\), where \(a_1\) is the first term and \(d\) is the common difference. Here, \(a_1 = 7\) and \(d = 2\). Substitute these values into the formula to find the equation: \(a_n = 7 + (n-1) imes 2\).

Step 3: Simplify the Equation

Expand and simplify the equation from Step 2. \(a_n = 7 + 2(n-1)\) becomes \(a_n = 7 + 2n - 2\) and then simplifies to \(a_n = 2n + 5\). This is the equation for the nth term of the sequence.

Step 4: Find the 33rd Term

Using the equation \(a_n = 2n + 5\), substitute \(n = 33\) to find the 33rd term: \(a_{33} = 2(33) + 5\). Calculate \(2(33) = 66\), and adding 5 gives \(71\). Thus, the 33rd term is 71.

Key Concepts

nth term formula

In arithmetic sequences, each term is derived from the one before it using a consistent rule. To find any term in this sequence, especially when it's not explicitly written out, we use the nth term formula. This formula is the mathematical equation that lets us calculate any term's value directly by its position in the sequence. When you have the first term, denoted as \( a_1 \), and the common difference, denoted as \( d \), you can find the nth term by plugging them into the formula:

• \( a_n = a_1 + (n-1) \times d \)

Here:- \( a_1 \) is the first term of the sequence.- \( d \) is the common difference.- \( n \) is the term number you wish to find.To make it simpler, just substitute \( a_1 \) and \( d \) into the formula to get the expression specific to your sequence. For instance, in our sequence, \( a_1 = 7 \) and \( d = 2 \), so the nth term formula becomes \( a_n = 7 + (n-1) \times 2 \). Simplifying, we get \( a_n = 2n + 5 \). This lets us find any term in the sequence just by plugging in the term number \( n \).

common difference

In the context of arithmetic sequences, the common difference is a key element. It describes how much we add to any term to reach the next one. This value remains the same throughout the sequence, emphasizing the regular pattern that defines arithmetic sequences. Understanding the common difference helps to quickly identify the nature of the sequence.
For our sequence, the terms are 7, 9, 11, and 13. We calculate the common difference by subtracting any term by its preceding term:

• \( 9 - 7 = 2 \)
• \( 11 - 9 = 2 \)
• \( 13 - 11 = 2 \)

Here, the common difference \( d \) is 2. This means each term increases by 2 from the previous term, confirming it’s an arithmetic sequence. Once the common difference is known, writing the nth term formula becomes an easier task, leading us back to the efficient expression, \( a_n = 2n + 5 \), ready for further exploration of terms.

sequence pattern

The sequence pattern is the underlying structure that defines how each term is related to the previous ones. In arithmetic sequences, this pattern is simple yet consistent. The consistent addition of the common difference is what creates this predictable pattern.
When starting from a known first term and understanding the common difference:

• Each term of the sequence shows how the pattern progresses.
• For example, beginning with 7 and consistently adding the common difference (2) produces the terms 9, 11, and 13 sequentially.

Using the sequence pattern, we can predict subsequent terms or find any term no matter how far along in the sequence. Linking back to the nth-term formula, \( a_n = 2n + 5 \), this expression encapsulates the sequence pattern, allowing direct computation of any term. The ease of this approach shows that grasping the sequence pattern saves time and reduces the potential for computation errors.

prealgebra

Prealgebra is a crucial stepping stone in mathematics, laying the foundation for algebra and beyond. It encompasses basic mathematical principles and introduces students to the concepts of variables and simple equations. A common topic tackled at this level is arithmetic sequences.
Studying sequences involves understanding how numbers can be ordered based on specific rules, such as the one featured in arithmetic sequences. The basics of prealgebra develop skills in:

• Recognizing patterns and differences between numbers.
• Solving equations to find unknown variables, in this case, specific terms in a sequence.
• Practicing the manipulation of equations like the nth term formula, \( a_n = a_1 + (n-1) \times d \).

Such exercises foster analytical thinking and problem-solving abilities. As learners master these prealgebra concepts, they pave the way for more advanced algebraic topics, building confidence and proficiency in handling mathematical challenges.