Finding The 24th Term In Arithmetic Sequence A Step-by-Step Guide
In the realm of mathematics, arithmetic sequences hold a fundamental position, characterized by a consistent difference between consecutive terms. This article delves into the process of identifying a specific term within such a sequence, focusing on the determination of the 24th term in the given sequence: -21, -14, -7, 0, 7, ...
Understanding Arithmetic Sequences
Before we embark on the calculation, let's solidify our understanding of arithmetic sequences. An arithmetic sequence is a series of numbers where the difference between any two successive terms remains constant. This constant difference is known as the common difference, often denoted by 'd'.
In our sequence, -21, -14, -7, 0, 7, ..., we can readily observe that the common difference is 7. Each term is obtained by adding 7 to its preceding term. This consistent pattern is the hallmark of an arithmetic sequence.
The general formula for the nth term (an) of an arithmetic sequence is given by:
an = a1 + (n - 1)d
Where:
- an represents the nth term
- a1 signifies the first term
- n denotes the term number
- d stands for the common difference
Identifying the First Term and Common Difference
To apply the formula effectively, we must first identify the first term (a1) and the common difference (d) from the given sequence.
In our sequence, -21, -14, -7, 0, 7, ..., the first term, a1, is clearly -21. This is the starting point of our sequence.
The common difference, d, can be calculated by subtracting any term from its subsequent term. For instance, subtracting -14 from -21 yields 7, and subtracting -7 from -14 also results in 7. This consistency confirms that our common difference, d, is indeed 7.
Calculating the 24th Term
Now, with the first term (a1 = -21) and the common difference (d = 7) firmly established, we can proceed to calculate the 24th term (a24) using the formula:
a24 = a1 + (24 - 1)d
Substituting the values we have:
a24 = -21 + (23)7
a24 = -21 + 161
a24 = 140
Therefore, the 24th term in the arithmetic sequence -21, -14, -7, 0, 7, ... is 140. This means that if we were to continue this sequence, the 24th number in the series would be 140. This calculation demonstrates the power of the arithmetic sequence formula in predicting terms far down the line without having to manually calculate each preceding term. The formula provides a direct and efficient method for finding any term in the sequence, given the first term, the common difference, and the desired term number.
Practical Applications of Arithmetic Sequences
Arithmetic sequences are not merely abstract mathematical constructs; they manifest in various real-world scenarios. Understanding them allows us to model and predict patterns in diverse situations.
Simple Interest: Consider a scenario where you deposit a certain sum of money into an account that earns simple interest. The interest earned each year remains constant, forming an arithmetic sequence. For instance, if you deposit $1000 and earn $50 in interest each year, the total amount in your account will increase by $50 annually, creating an arithmetic sequence.
Depreciation: In the realm of finance, depreciation refers to the decrease in the value of an asset over time. If an asset depreciates by a fixed amount each year, its value forms an arithmetic sequence. For example, if a car's value decreases by $2000 each year, its value will follow an arithmetic progression.
Patterns in Nature: Arithmetic sequences can also be observed in natural phenomena. The number of petals in certain types of flowers, the arrangement of leaves on a stem, and the segments in a pineapple often follow arithmetic patterns. These patterns reveal the underlying mathematical order in the natural world.
Construction and Engineering: In construction and engineering, arithmetic sequences are used in designing structures, calculating materials needed, and managing project timelines. For instance, the spacing of bricks in a wall or the number of steps in a staircase can be determined using arithmetic sequences.
Common Mistakes to Avoid
When working with arithmetic sequences, it's essential to be mindful of potential pitfalls that can lead to errors. Here are some common mistakes to avoid:
Incorrectly Identifying the First Term: The first term (a1) is the starting point of the sequence. Ensure you correctly identify it, as an error here will propagate through subsequent calculations.
Miscalculating the Common Difference: The common difference (d) is the constant value added to each term to obtain the next. Accurately determine 'd' by subtracting any term from its subsequent term. Be consistent in your calculations to avoid mistakes.
Applying the Formula Incorrectly: The formula for the nth term (an = a1 + (n - 1)d) is the cornerstone of arithmetic sequence calculations. Ensure you substitute the values correctly, paying close attention to the order of operations.
Confusion with Geometric Sequences: Arithmetic sequences involve a constant difference, while geometric sequences involve a constant ratio. Be mindful of this distinction and avoid applying the wrong formula.
Not Checking Your Answer: After calculating the nth term, it's prudent to check your answer. You can do this by manually calculating a few terms around the nth term to see if your result aligns with the sequence pattern. This simple step can help you identify and rectify errors.
Conclusion
In conclusion, finding the 24th term in the arithmetic sequence -21, -14, -7, 0, 7, ... involves a systematic application of the arithmetic sequence formula. By accurately identifying the first term, common difference, and term number, we can efficiently determine the value of any term in the sequence. Understanding arithmetic sequences provides a valuable tool for modeling patterns and making predictions in various mathematical and real-world contexts. Remember to avoid common pitfalls and always double-check your calculations to ensure accuracy.