Problem 28 - College Algebra

Question

Write a formula for the general term (the nth term) of each arithmetic sequence. Do not use a recursion formula. Then use the formula for \(a_{n}\) to find \(a_{20}\), the 20 th term of the sequence. $$a_{1}=6, d=3$$

Step-by-Step Solution

Answer

The formula for the nth term of the sequence is \(a_{n} = 3n + 3\) and the 20th term of the sequence is 63.

Step 1: Write down given values

Firstly, note down the values which are given:The first term or \(a_{1} = 6\) and the common difference of the arithmetic sequence \(d = 3\).

Step 2: Derive the formula for general term

By using the formula for an arithmetic progression, we can express the \(n^{th}\) term (\(a_{n}\)) as follows: \(a_{n} = a_{1} + (n-1) * d\). In this case, we substitute \(a_{1} = 6\) and \(d = 3\) to get \(a_{n} = 6 + (n-1) * 3\). This can be simplified further to \(a_{n} = 3n + 3\). This is the formula for the \(n^{th}\) term of the arithmetic sequence.

Step 3: Find the 20th term

Plug in the value \(n = 20\) into our formula to find the 20th term. So, \(a_{20} = 3*20 + 3 = 63\).

Key Concepts

Understanding the General Term Formula for Arithmetic Sequences

In arithmetic sequences, every term after the first is derived by adding a fixed number, called the 'common difference', to the preceding term. To easily find any term in this sequence, we use the general term formula, which provides a way to calculate the nth term, denoted as \(a_{n}\), without computing all the prior terms.

This formula is represented as:

• \(a_{n} = a_{1} + (n-1) \cdot d\)

• \(a_{1}\) is the first term of the sequence.
• \(d\) is the common difference between terms.
• \(n\) refers to the term number you're interested in finding.

For example, if your sequence starts at 6 (\(a_{1} = 6\)) and has a common difference of 3 (\(d = 3\)), the formula allows a straightforward computation of any term. Substituting these values into the formula gives \(a_{n} = 6 + (n-1) \cdot 3\).
This formula simplifies calculating terms rather than progressing step by step through the sequence.

Decoding the Common Difference in Arithmetic Sequences

The common difference is a crucial element of an arithmetic sequence. It is the constant value added to each term to get the next. Understanding this difference helps to easily generate the sequence and to calculate any specific term using the general term formula.

Consider the sequence given in the exercise: it has a first term of 6 and a common difference of 3. This tells us that each term is constructed by adding 3 to the preceding term. The sequence starts as 6, 9, 12, 15, ... and so on.

• Calculating the common difference is as simple as performing subtraction: take any term and subtract its preceding term, for example, \(d = 9 - 6 = 3\).
• This subtractive property assists in verifying or determining the consistency of the sequence.

With the common difference at hand, forming the general term becomes an approachable task. It enables efficient construction and computation of the sequence’s terms.

Efficient nth Term Calculation with Included Example

Once the general term formula is established, calculating any nth term becomes a straightforward task. The exercise asks for the 20th term of the series, which is computed using the derived formula from the common difference and initial term.

• Plug \(n = 20\) into the general term formula: \(a_{n} = 6 + (n-1) \cdot 3\).
• This translates to \(a_{20} = 6 + 19 \cdot 3\).
• By performing the arithmetic operation, \(a_{20} = 6 + 57 = 63\).

Thus, the 20th term of the sequence is 63.
This calculation is simplified because of the earlier discovery of the common difference and the standard formula. Tackling sequences with such consistency allows for rapid computation of potentially any term, provided the sequence characteristics are well-defined.