Problem 25 - Prealgebra

Question

Write an equation that describes each sequence. Then find the indicated term. \(11,22,33,44, \dots ; 25\) th term

Step-by-Step Solution

Answer

The equation is \(a_n = 11n\). The 25th term is 275.

Step 1: Identify the Sequence Type

The given sequence is an arithmetic sequence because the difference between consecutive terms is constant. Here, the difference between each term is 11. For example, \(22 - 11 = 11\), \(33 - 22 = 11\).

Step 2: Write the General Formula for the Arithmetic Sequence

The general formula for an arithmetic sequence is given by \(a_n = a_1 + (n-1)d\), where \(a_1\) is the first term, \(d\) is the common difference, and \(n\) is the term number.

Step 3: Substitute Known Values into the Formula

Here, \(a_1 = 11\) and \(d = 11\). Substitute these values into the formula to get the equation of the sequence: \(a_n = 11 + (n-1) \times 11\).

Step 4: Simplify the Equation

Simplify the equation: \(a_n = 11 + 11(n-1) = 11 + 11n - 11 = 11n\). So, the nth term of the sequence is described by \(a_n = 11n\).

Step 5: Find the 25th Term

Substitute \(n = 25\) into the equation \(a_n = 11n\) to find the 25th term: \(a_{25} = 11 \times 25 = 275\).

Key Concepts

General Formula for Arithmetic Sequence

In mathematics, an arithmetic sequence is a series of numbers in which the difference between any two successive terms is constant. This constant difference is known as the "common difference." The general formula used to represent the nth term of an arithmetic sequence is:

\[a_n = a_1 + (n-1)d\]Where:

• \(a_n\) is the nth term of the sequence.
• \(a_1\) is the first term of the sequence.
• \(d\) is the common difference.
• \(n\) represents the term number we are trying to find.

This formula allows us to calculate any term in the sequence without having to write down all the preceding terms. It is a powerful tool for simplifying problems involving arithmetic sequences.

Common Difference in Sequences

The "common difference" is a fundamental concept in arithmetic sequences. It is the amount added to each term to get the next term. For example, in the sequence given in the exercise \(11, 22, 33, 44, \dots\), the common difference is:

\[22 - 11 = 33 - 22 = 11\]This common difference \(d\) is what makes the sequence an arithmetic sequence. It remains the same across the entire sequence. Understanding how to find and use this difference is critical when working with arithmetic sequences as it directly influences the general formula and helps in finding specific terms within the sequence.

Finding nth Term in a Sequence

Finding the nth term in an arithmetic sequence is straightforward once you know the first term and the common difference. By substituting into the general formula, you can obtain any term you need. For instance, in the provided sequence where:

• First term \(a_1 = 11\)
• Common difference \(d = 11\)

You can find the 25th term by substituting \(n = 25\) into the general formula:

\[a_{25} = 11 + (25-1) \times 11 = 11 + 24 \times 11\]This simplifies to:

\[a_{25} = 11 + 264 = 275\]So, the 25th term of the sequence is 275. This method can be applied to any arithmetic sequence, making it a vital skill for solving sequence-related problems.