Exploring Arithmetic Sequences: Finding a Term That is 72 More Than the 41st Term

Understanding and analyzing arithmetic sequences can be a fascinating challenge. In this article, we will explore how to find a term in an arithmetic sequence that is 72 more than the 41st term. We will also see the practical applications and methods involved in solving such problems.

Introduction to Arithmetic Sequences

An arithmetic sequence is a sequence of numbers in which the difference between any two successive members is a constant. For instance, the arithmetic sequence 8, 14, 20, 26 has a first term a 8 and a common difference d 6.

Identifying the 41st Term in the Sequence

Let's start by identifying the 41st term in the sequence. The formula for the nth term of an arithmetic sequence is given by:

Tn a (n - 1) d

For the 41st term:

T41 8 (41 - 1) * 6 8 40 * 6 8 240 248

Identifying the Term That is 72 More Than the 41st Term

The problem asks for a term in the sequence that is 72 more than the 41st term. So we need to find the term that equals 248 72, which is:

248 72 320

We now need to find which term in the sequence equals 320. Using the general formula for the nth term:

Tn 8 (n - 1) * 6 320

Solving for n gives:

(n - 1) * 6 320 - 8 312

n - 1 312 / 6 52

n 52 1 53

Therefore, the 53rd term is 320.

Summary and Conclusion

In this article, we've explored the process of finding a term in an arithmetic sequence that is 72 more than the 41st term. We achieved this by using the general formula of the nth term of an arithmetic sequence, followed by solving algebraic equations. The conclusion indicates that the term we are looking for is indeed 320, which is the 53rd term in the given sequence.

Additional Resources and Practice

If you need further practice or a more detailed understanding, I recommend checking out the following resources:

Video Tutorial on Arithmetic Sequences Khan Academy - Finding the nth Term of an Arithmetic Sequence

Remember, practice is key to mastering these concepts. The resources provided can help you reinforce your understanding and prepare for similar problems in the future.

Happy Learning!