Problem 9 - College Algebra

Question

In Exercises \(1-14\), write the first six terms of cach arithmetic sequence $$a_{n}=a_{n-1}+6, a_{1}=-9$$

Step-by-Step Solution

Answer

The first six terms of the given arithmetic sequence are -9, -3, 3, 9, 15, 21.

Step 1: First Term

The first term of the arithmetic sequence is given as -9. So, write \(a_{1}=-9\).

Step 2: Second Term

To find the second term, use the formula \(a_{n}=a_{n-1}+6\), where \(n=2\) and \(a_{n-1}=a_{1}\). This gives \(a_{2}=a_{1}+6 = -9+6 = -3\).

Step 3: Third Term

Repeat Step 2 with \(n=3\) and \(a_{n-1}=a_{2}\) to get \(a_{3}=a_{2}+6 = -3+6 = 3\).

Step 4: Fourth Term

Same as in step 3 but \(n=4\) and \(a_{n-1}=a_{3}\), It gives \(a_{4}=a_{3}+6 = 3+6 = 9\).

Step 5: Fifth Term

Continuing as before, for \(n=5\) and \(a_{n-1}=a_{4}\), compute \(a_{5}=a_{4}+6 = 9+6 = 15\).

Step 6: Sixth Term

Repeat step again for \(n=6\) and \(a_{n-1}=a_{5}\), hence the sixth term \(a_{6}=a_{5}+6 = 15+6 = 21\).

Key Concepts

Sequence Terms

An arithmetic sequence is a number series where each term after the first is found by adding a fixed number to the previous term. This fixed number is known as the common difference. In the given exercise, the sequence starts with

• First term: \(a_1 = -9\)

To determine the sequence terms, this common difference is added repeatedly. The sequence generated by these steps is often written out as

• -9,
• -3,
• 3,
• 9,
• 15,
• 21, ...

Understanding the setup with sequence terms is crucial to grasp the full arithmetic sequence.

Term Addition

Term addition is a method used to find each term in the sequence by adding the common difference to the previous term. In this context, the common difference is 6. This means every time you calculate a new term, you increase the current value by 6.

• For example, starting from -9, you add 6 to get the next term.
• This step is repeated for each subsequent term. Each addition involves simple arithmetic.

Term addition is a fundamental step in constructing and understanding arithmetic sequences, and it allows sequences to extend indefinitely in a predictable pattern.

Step-by-Step Calculation

A step-by-step calculation helps break down the process of finding sequence terms into straightforward, manageable parts. Here is how you can use it:

• Begin with the known first term, \(a_1 = -9\).
• Apply the term addition repeatedly using the sequence formula given for each new term.
• Specifically, calculate the next term as follows: \(a_n = a_{n-1} + 6\).

This method enhances comprehension, as it logically sequences the arithmetic steps involved, making the sequence development intuitive and accessible.

Sequence Formula

The sequence formula is a mathematical expression used to define the rule of the arithmetic sequence. The formula given \(a_n = a_{n-1} + 6\) specifies that each term (\(a_n\)) can be found by adding 6 to the preceding term (\(a_{n-1}\)).

• This recursive formula signifies the arithmetic nature, whereby a fixed increment leads to subsequent terms.

Understanding this formula is essential, as it allows calculation of any term in the sequence from the preceding one without needing to recount all previous terms. This is both efficient and crucial for dealing with large sequences. A key takeaway is the constant 6 in the formula, which maintains the uniform increment characteristic of the arithmetic sequence.