Problem 3 - Algebra and Trigonometry

Question

Write the first six terms of each arithmetic sequence. $$ a_{1}=-7, d=4 $$

Step-by-Step Solution

Answer

The first six terms of this arithmetic sequence are -7, -3, 1, 5, 9, 13.

Step 1: Calculate the First term

We are given that the first term \(a_{1}\) is -7, so, the first term of the sequence is -7.

Step 2: Calculate the Second term

To find the second term we substitute n=2 in the arithmetic sequence formula: \(a_{2} = a_{1} + (2-1)*d = -7 + 4 = -3\). Hence, the second term is -3.

Step 3: Calculate the Third term

Substitute n=3 in the formula: \(a_{3} = a_{1} + (3-1)*d = -7 + 4*2 = 1\). So, the third term is 1.

Step 4: Calculate the Fourth term

Substitute n=4 in the formula: \(a_{4} = a_{1} + (4-1)*d = -7 + 4*3 = 5\). So, the fourth term is 5.

Step 5: Calculate the Fifth term

Now, substitute n=5 into the formula: \(a_{5} = a_{1} + (5-1)*d = -7 + 4*4 = 9\). The fifth term of the sequence is 9.

Step 6: Calculate the Sixth term

Finally, substitute n=6 into the formula: \(a_{6} = a_{1} + (6-1)*d = -7 + 4*5 = 13\). The sixth term of the sequence is 13.

Key Concepts

Sequence Terms

In arithmetic sequences, we explore ordered lists of numbers where each term follows a specific pattern. Each number in this list is called a 'term'.
The terms in an arithmetic sequence are generated based off a fixed starting point, known as the first term, and a consistent amount added to each subsequent term, which is called the common difference. An arithmetic sequence continues infinitely, unless otherwise specified, and is defined by these two components:

• First Term (\(a_{1}\))
• Common Difference (\(d\))

Let's illustrate this with our example: starting with the first term, which is -7, and using a common difference of 4, the progression of sequence terms can be systematically created.

Common Difference

A key feature of arithmetic sequences is the 'common difference'. It's the amount that separates one term from the next, consistently throughout the sequence.
In our example, the common difference is given as 4. This implies that you add 4 to the value of the current term to find the next term in the sequence.

• If the common difference is positive, the sequence increases.
• If it's negative, the sequence decreases.

Understanding how to apply the common difference allows you to predict and calculate any term in the sequence, whether you're moving forward or backward through the sequence.

Arithmetic Progression

Arithmetic progression is simply another term for arithmetic sequence. It outlines the systematic approach to developing the sequence's terms by using the first term and the common difference.
Such sequences are linear and can be represented generally as:

• \(a_{1}\) is the first term,
• \(a_{2} = a_{1} + d\)
• And so on...

These terms grow in a predictable pattern, and this type of repetition is called progression. In our example, the sequence begins at -7 and grows by adding 4 repeatedly, making it straightforward and easy to follow.

Term Calculation

The calculation of terms in an arithmetic sequence is an application of a simple formula:\[a_{n} = a_{1} + (n-1) imes d\]This formula is quite powerful because it allows us to find any term directly.
In this formula:

• \(a_{n}\) stands for the term we want to find,
• \(a_{1}\) is the known first term,
• \(n\) is the position of the term,
• \(d\) is the common difference.

Using this method, we've calculated the first six terms in our example: -7, -3, 1, 5, 9, and 13. Each term can be quickly identified using the formula rather than manual counting, enabling efficiency and accuracy in finding terms.