Problem 58 - Algebra 2

Question

Find the indicated term of each arithmetic sequence. \(a_{21}\) for \(121,118,115, \ldots\)

Step-by-Step Solution

Answer

The 21st term, \(a_{21}\), is 61.

Step 1: Identify the First Term and Common Difference

The first term of the sequence is \(a_1 = 121\). The common difference \(d\) is found by subtracting the second term from the first term: \(118 - 121 = -3\). Thus, \(d = -3\).

Step 2: Use the Formula for the n-th Term of an Arithmetic Sequence

The formula to find the n-th term of an arithmetic sequence is given by \(a_n = a_1 + (n-1) \cdot d\). Here, we need to find \(a_{21}\), so we will substitute the known values into this formula.

Step 3: Substitute Values into the Formula

We need to find \(a_{21}\). Using the formula \(a_n = a_1 + (n-1) \cdot d\), substitute \(a_1 = 121\), \(d = -3\), and \(n = 21\). This gives us: \[ a_{21} = 121 + (21-1) \cdot (-3) \]

Step 4: Simplify the Expression

First, calculate \(21-1\) which is \(20\). Then multiply \(20\) by the common difference \(-3\), resulting in \(20 \cdot (-3) = -60\). Substitute this back into the expression: \[ a_{21} = 121 - 60 \]

Step 5: Calculate the Final Result

Perform the subtraction \(121 - 60\) to find the 21st term. Thus, \(a_{21} = 61\).

Key Concepts

Common Difference

In an arithmetic sequence, the common difference represents the consistent interval that separates consecutive terms in the sequence. To determine this value, subtract any term from the subsequent term in the sequence.
For example, in sequence 121, 118, 115, and so forth, the common difference is calculated as 118 - 121, resulting in -3.
Key points about the common difference include:

• If the difference is positive, the sequence will increase with each term.
• If negative, as in our example -3, the sequence will decrease.
• A common difference of 0 results in a sequence where all terms are identical.

The consistent quality of the common difference drives the entire pattern of arithmetic sequences, making it essential to identify it accurately.

First Term

The first term of an arithmetic sequence is the point from which all calculations start. It's the initial member of the sequence, frequently denoted as \(a_1\).
In our sequence of 121, 118, 115, etc., the first term \(a_1\) is 121.
Understanding the first term is crucial because it acts like the anchor for the rest of the sequence. Without it, determining other terms using the formula is impossible.
In arithmetic sequences:

• This term is necessary to identify the entire sequence.
• It provides the starting value from which all subsequent terms are derived using the common difference.

Mastery of the first term ensures you can use it in conjunction with the common difference to find any term in the sequence.

n-th Term Formula

The n-th term formula in an arithmetic sequence allows you to find any specified term without listing all preceding terms. It's given by \(a_n = a_1 + (n-1) \cdot d\), where:

• \(a_n\) is the term you aim to find.
• \(a_1\) is the first term.
• \(n\) represents the term number.
• \(d\) is the common difference between terms.

Suppose you want to find \(a_{21}\) of our sequence, with \(a_1 = 121\), \(d = -3\), and \(n = 21\). Substitute these values into the formula:\[a_{21} = 121 + (21-1) \cdot (-3) \]This leads to calculating \(20 \cdot (-3) = -60\), which results in the 21st term: \(61\).
Overall, this formula is a powerful tool in arithmetic sequences, minimizing manual effort and swiftly obtaining outcomes.