Finding The 35th Term Of An Arithmetic Sequence With A1 = -7 And A18 = 95
In the realm of mathematics, arithmetic sequences hold a fundamental position, serving as the bedrock for various advanced concepts. An arithmetic sequence is characterized by a constant difference between consecutive terms, a property that allows us to predict any term in the sequence with precision. This article delves into the process of identifying the 35th term of an arithmetic sequence, given the first term ($a_1$) and the 18th term ($a_{18}$). We will explore the underlying principles, derive the necessary formulas, and apply them to solve the problem at hand. Whether you are a student grappling with arithmetic sequences or a seasoned mathematician seeking a refresher, this guide will provide you with a clear and concise understanding of the topic.
Understanding Arithmetic Sequences
At its core, an arithmetic sequence is a series of numbers where the difference between any two consecutive terms remains constant. This constant difference is known as the common difference, often denoted by the letter 'd'. The sequence progresses by adding or subtracting this common difference to each preceding term. For instance, the sequence 2, 5, 8, 11, 14... is an arithmetic sequence with a common difference of 3.
To fully grasp arithmetic sequences, it is essential to understand the notation and terminology used to describe them. The first term of the sequence is typically denoted as $a_1$, the second term as $a_2$, and so on. The nth term of the sequence is represented as $a_n$. The common difference, as mentioned earlier, is denoted as 'd'.
The general formula for the nth term of an arithmetic sequence is given by:
where:
-
a_n$ is the nth term of the sequence
-
a_1$ is the first term of the sequence
- n is the term number
- d is the common difference
This formula forms the cornerstone of our analysis, allowing us to calculate any term in the sequence if we know the first term and the common difference. Let's delve deeper into how we can determine these crucial parameters.
Determining the Common Difference (d)
The common difference is the linchpin of an arithmetic sequence, dictating how the sequence progresses. To find the common difference, we simply subtract any term from its succeeding term. In other words:
d = a_2 - a_1 = a_3 - a_2 = a_4 - a_3$ and so on. However, in many scenarios, we may not have consecutive terms readily available. Instead, we might be given two non-consecutive terms, such as the first term and a later term. In such cases, we can adapt the general formula to determine the common difference. Suppose we know the mth term ($a_m$) and the nth term ($a_n$) of an arithmetic sequence, where m and n are different term numbers. We can write the following equations using the general formula: $a_m = a_1 + (m - 1)d
Subtracting the first equation from the second, we get:
Solving for d, we obtain the formula for the common difference:
d = rac{a_n - a_m}{n - m}
This formula allows us to calculate the common difference even when we are not provided with consecutive terms, expanding our ability to analyze arithmetic sequences.
Identifying the 35th Term
Now that we have established the fundamental principles and formulas, let's tackle the problem at hand: identifying the 35th term of an arithmetic sequence where $a_1 = -7$ and $a_{18} = 95$.
Our first step is to determine the common difference (d). We are given two terms, $a_1$ and $a_{18}$, so we can use the formula we derived earlier:
d = rac{a_n - a_m}{n - m}
In this case, $a_n = a_{18} = 95$, $a_m = a_1 = -7$, n = 18, and m = 1. Plugging these values into the formula, we get:
d = rac{95 - (-7)}{18 - 1} = rac{102}{17} = 6
Therefore, the common difference of this arithmetic sequence is 6. Now that we know the first term and the common difference, we can use the general formula to find the 35th term:
We want to find $a_{35}$, so we set n = 35, $a_1 = -7$, and d = 6:
Thus, the 35th term of the arithmetic sequence is 197.
Step-by-Step Solution
To summarize, here's a step-by-step breakdown of how to find the 35th term of the arithmetic sequence:
- Identify the given information: We are given $a_1 = -7$ and $a_{18} = 95$.
- Calculate the common difference (d): Use the formula $d = rac{a_n - a_m}{n - m}$ with $a_{18}$ and $a_1$ to find d. $d = rac{95 - (-7)}{18 - 1} = 6$.
- Apply the general formula: Use the formula $a_n = a_1 + (n - 1)d$ to find $a_{35}$. $a_{35} = -7 + (35 - 1)6 = 197$.
Conclusion
In this comprehensive guide, we have explored the concept of arithmetic sequences and learned how to identify any term in the sequence, given sufficient information. We derived the formulas for the common difference and the nth term, and applied them to find the 35th term of a specific arithmetic sequence. By understanding the underlying principles and mastering these formulas, you can confidently tackle any problem involving arithmetic sequences. Remember, practice is key to solidifying your understanding, so don't hesitate to work through various examples and exercises. Arithmetic sequences are a fundamental building block in mathematics, and a strong grasp of this concept will undoubtedly benefit you in your further mathematical endeavors.
- Arithmetic sequence
- Common difference
- nth term
- Formula for arithmetic sequence
- Finding the 35th term
- Mathematics
- Sequence and series
- $a_1$
- $a_{18}$
- $a_{35}$
