Unveiling the 47th Term of the Arithmetic Sequence

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 1/12

Substituting 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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