Unveiling the 47th Term of the Arithmetic Sequence
In this article, we will delve into the details of an essential concept in mathematics: the arithmetic sequence. We will specifically explore how to find the 47th term of an arithmetic sequence where the first three terms are 1/6, 1/3, and we will determine the difference to understand the sequence better.
Identifying the Differences in the Sequence
Given an arithmetic sequence with the first term a 1/6, to find the common difference d, we calculate the difference between the second and first term. By subtracting the first term (1/6) from the second term (1/3), we get:
d 1/3 - 1/6 2/6 - 1/6 1/6.
Determining the n-th Term of an Arithmetic Sequence
The n-th term of an arithmetic sequence (Tn) can be given by the general formula:
Tn a (n - 1)d.
Here, 1/3 is known to be the second term, and we need to find which term is 4. Let's denote the term 4 as Tn. Applying the formula:
4 1/6 (n - 1)(1/12)
4 - 1/6 n - 1(1/12)
24/6 - 1/6 n - 1(1/12)
23/6 n - 1(1/12)
23/6 * 12 n - 1
23 * 2 n - 1
46 n - 1
n 46 1
n 47
Hence, the 47th term of this arithmetic sequence is 4.
Visualizing the Solution
Let's break down the n-th term formula to visualize the process:
an a (n - 1)d
Given:
an 4 a 1/6 d 1/12Substituting in the formula:
4 1/6 (n - 1)(1/12)
To isolate n - 1, rearrange:
4 - 1/6 (n - 1)(1/12)
24/6 - 2/12 (n - 1)(1/12)
23/6 (n - 1)(1/12)
Multiplying both sides by 12:
23 * 2 n - 1
46 n - 1
n 47
Therefore, the 47th term is indeed 4.
Conclusion
Understanding the concept of arithmetic sequences and the formula for finding the n-th term is crucial. The above solution illustrates the step-by-step process to find the term 4 in the given sequence. There are various online resources, such as math websites, which offer detailed explanations and interactive tools to help visualize and solve similar problems.
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