Problem 2 - Algebra and Trigonometry

Question

Write the first six terms of each arithmetic sequence. $$ a_{1}=300, d=50 $$

Step-by-Step Solution

Answer

The first six terms of the arithmetic sequence are 300, 350, 400, 450, 500, 550.

Step 1: Write down the first term

The first term of the sequence, denoted as \(a_{1}\), has been given as 300.

Step 2: Add the common difference to the first term

To find the second term, add the common difference, which is 50, to the first term. So, \(a_{2} = a_{1} + d = 300 + 50 = 350\).

Step 3: Continue the process for the rest of the terms

Keep on adding the common difference to the preceding term to get the next term for as long as needed. Now, \(a_{3} = a_{2} + d = 350 + 50 = 400\), \(a_{4} = a_{3} + d = 400 + 50 = 450\), \(a_{5} = a_{4} + d = 450 + 50 = 500\), \(a_{6} = a_{5} + d = 500 + 50 = 550\).

Key Concepts

Arithmetic Progression

An arithmetic progression (AP) is a sequence of numbers in which each term after the first is generated by adding a constant value, known as the common difference, to the previous term. The AP is a way to represent a pattern of numbers that has a natural rhythm or predictability.

For example, consider the arithmetic progression that begins with a first term, or initial value, of 300. If we continue this sequence by adding the same fixed amount each time, we create a list of numbers in a specific order. This characteristic of an arithmetic progression allows us to find any term in the sequence by applying a simple formula, not just by counting or adding repeatedly.

Common Difference

The 'common difference' is a key element in an arithmetic progression. It is the steady amount by which consecutive terms increase (or decrease, if it is negative) as you move from one term to the next in the sequence.

To identify the common difference in a given AP, you subtract one term from the subsequent term. For instance, with the provided exercise, the common difference is 50 because each term is 50 units larger than the term before it. Once the common difference is known, calculating the rest of the terms becomes a matter of simple repetitive addition. Understanding this concept is critical because it underpins the entire structure of arithmetic sequences.

Sequence Terms

The individual members of an arithmetic progression are called 'sequence terms'. These terms are often denoted by the symbol 'a' followed by a subscript indicating its position in the sequence, such as \( a_{1} \) for the first term, \( a_{2} \) for the second term, and so forth.

When working with sequence terms, the position of a term can be crucial information. Not only does it help us refer to specific elements in the sequence, but it also aids in mathematical operations like finding the nth term using the formula \( a_{n} = a_{1} + (n-1)d \), where 'n' represents the term's position in the sequence. Each term can be methodically determined by adding the common difference repeatedly from the first term, or directly by using the formula that encapsulates the entire concept. This structured approach ensures a clear and comprehensible way to navigate through arithmetic sequences.