Problem 2 - Algebra and Trigonometry

Question

The sequence \(a_{n}=a+(n-1) d\) is an arithmetic sequence in which a is the first term and \(d\) is the ________ __________ So for the arithmetic sequence \(a_{n}=2+5(n-1)\) the first term is ___________and the common difference is _____________

Step-by-Step Solution

Answer

First term: 2, common difference: 5.

Step 1: Identify the General Form

An arithmetic sequence has the general formula \( a_n = a + (n-1)d \), where \( a \) is the first term and \( d \) is the common difference.

Step 2: Match the Given Sequence to the General Form

The given arithmetic sequence is \( a_n = 2 + 5(n-1) \). This must be matched to the general formula \( a_n = a + (n-1)d \).

Step 3: Identify the First Term \(a\)

In the equation \( a_n = 2 + 5(n-1) \), compare with \( a_n = a + (n-1)d \) to find \( a = 2 \).

Step 4: Determine the Common Difference \(d\)

In the equation \( a_n = 2 + 5(n-1) \), compare with \( a_n = a + (n-1)d \) to find \( d = 5 \).

Key Concepts

Common Difference

In an arithmetic sequence, the term "common difference" refers to the constant amount that separates consecutive terms. This is a key feature of arithmetic sequences, where every term after the first is derived by adding the common difference to the previous term.
Understanding the common difference is essential because it allows us to predict future terms of the sequence. If you know the first term and the common difference, you can continue the sequence indefinitely.
For the arithmetic sequence given by the formula \( a_n = 2 + 5(n-1) \), the common difference is 5. This means that each term is 5 more than the preceding term.

• Example: If your first term is 2, the second term would be \( 2 + 5 = 7 \), the third term \( 7 + 5 = 12 \), and so on.

First Term

The first term of an arithmetic sequence, often represented by the symbol \( a \), is the starting point of the sequence. It sets the stage for all subsequent terms, as it is the initial value to which the common difference is added repeatedly.
Identifying the first term is crucial, as it provides the foundation from which the sequence is built.
In our specific sequence example \( a_n = 2 + 5(n-1) \), the first term \( a \) is clearly defined as 2.

• This means that as you calculate each subsequent term, you begin with 2 and add the common difference repeatedly.
• Knowing the first term allows you to reassess the sequence or backtrack when needed.

General Formula of Arithmetic Sequence

The general formula of an arithmetic sequence is a powerful tool in mathematics represented as \( a_n = a + (n-1)d \).
This formula is used to determine any term in the sequence without having to calculate all the preceding terms.
Here, \( a_n \) represents the \( n \)-th term in the sequence, \( a \) is the first term, \( n \) is the term number, and \( d \) is the common difference.

• This formula reveals how each term is constructed systematically from the first term by adding entire multiples of the common difference.
• In the case of \( a_n = 2 + 5(n-1) \), the formula tells us that the sequence starts from 2, and each term increases by a multiple of 5.

Using this formula, you can effortlessly find any desired term in the sequence, making it a versatile and indispensable mathematical tool.