Problem 15 - Algebra and Trigonometry

Question

Find the indicated term of the arithmetic sequence with first term, \(a_{1},\) and common difference, \(d .\) Find \(a_{6}\) when \(a_{1}=13, d=4\)

Step-by-Step Solution

Answer

So, the 6th term of the given arithmetic sequence is 33.

Step 1: Insert the Values into the Formula

Using the term-seeking formula for an arithmetic sequence, general term in arithmetic sequence is given by the formula: \(a_{n} = a_{1} + (n - 1) \cdot d\). Now input the known values: \(a_{1} = 13\), \(d = 4\) and \(n = 6\). So the formula becomes: \(a_{6} = 13 + (6 - 1) \cdot 4\).

Step 2: Calculating the difference Part

Initially, let's solve the section in the brackets: \(6 - 1 = 5\). Multiply this by the common difference which is 4, the equation becomes: \(a_{6} = 13 + 5 \cdot 4\).

Step 3: Calculate the Final Result

Finally, follow the order of operations to solve the right side of the equation: \(a_{6} = 13 + 20 = 33\).

Key Concepts

Understanding the First Term in Arithmetic Sequences

The first term of an arithmetic sequence is the starting point of the entire sequence. It is denoted as \(a_1\). This term sets the foundation for constructing the rest of the sequence.
For example, if your first term is given as 13, as in the exercise mentioned, it means that the sequence begins with this number.
The first term is crucial because every other term in the sequence builds upon it.

• It forms the basis from which the sequence unfolds.
• In the given exercise, since \(a_1 = 13\), all future terms are derived by adding specific multiples of the common difference \(d\) to this starting number.

This understanding of the first term allows us to consistently compute other terms in the sequence using the term formula, which will be discussed later.

Exploring the Common Difference

The common difference, often represented by \(d\), is a vital component of an arithmetic sequence. It indicates the amount by which each successive term increases as compared to the previous term.
In essence, the common difference tells us the consistent step size between terms.

• A positive common difference means each term is larger than the preceding one.
• A negative common difference implies that the terms decrease as the sequence progresses.

In the given problem, the common difference is 4. This means each term is 4 units more than the one before it. When combined with a first term of 13, the second term would be \(13 + 4 = 17\), the third \(17 + 4 = 21\), and so on.

Using the Term Formula to Find Any Term

The term formula in arithmetic sequences is a powerful tool that helps you determine any term in the sequence without calculating each preceding term. It is generally expressed as \(a_n = a_1 + (n - 1) \cdot d\), where:

• \(a_n\) is the term you wish to find.
• \(a_1\) is the first term of the sequence.
• \(d\) represents the common difference.
• \(n\) is the position of the term in the sequence.

To use the formula correctly, substitute the known values into \(a_n = a_1 + (n - 1) \cdot d\).
For the exercise we considered, finding the sixth term means setting \(n = 6\), \(a_1 = 13\), and \(d = 4\). This simplifies to \(a_6 = 13 + (6 - 1) \cdot 4\). Solving inside the brackets first, gives \(13 + 5 \cdot 4\), which results in 33. Hence, the sixth term is 33, illustrating how effective the term formula is.