The Infinite Number Sequence Below Is A Combination Of Two Sequences: $ 1, 1, 1, 5, 1, 9, 1, 13, \ldots\$} Determine 4.1.1 { T_{31 $}$, The Thirty-first Term Of The Sequence.4.1.2 { S_{32}$}$, The Sum Of The First

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Introduction

The infinite number sequence ${1, 1, 1, 5, 1, 9, 1, 13, \ldots\$} is a combination of two sequences. To determine the thirty-first term of the sequence, {T_{31}$}$, and the sum of the first thirty-two terms, {S_{32}$}$, we need to understand the underlying patterns of the sequence.

Understanding the Sequence

The given sequence can be broken down into two separate sequences: ${1, 1, 1, \ldots\$} and ${5, 9, 13, \ldots\$}. The first sequence consists of consecutive 1s, while the second sequence consists of consecutive odd numbers starting from 5.

Identifying the Pattern

To identify the pattern, let's analyze the sequence further. The second sequence can be represented as ${5, 9, 13, 17, 21, 25, 29, 33, \ldots\$}. We can observe that each term in the second sequence is obtained by adding 4 to the previous term.

Deriving the Formula

Based on the pattern, we can derive a formula for the nth term of the second sequence. Let's denote the nth term as {a_n$}$. We can write the formula as:

{a_n = 4n + 1$}$

Calculating the Thirty-First Term

Now that we have the formula for the nth term, we can calculate the thirty-first term of the sequence, {T_{31}$}$. We can substitute n = 31 into the formula:

{T_{31} = 4(31) + 1 = 124 + 1 = 125$}$

Calculating the Sum of the First Thirty-Two Terms

To calculate the sum of the first thirty-two terms, {S_{32}$}$, we need to consider both sequences. The first sequence consists of thirty-two 1s, while the second sequence consists of the first thirty-two terms of the sequence ${5, 9, 13, 17, 21, 25, 29, 33, \ldots\$}.

Deriving the Formula for the Sum

We can derive a formula for the sum of the first n terms of the second sequence. Let's denote the sum as {S_n$}$. We can write the formula as:

{S_n = \frac{n(n + 1)}{2} \times 4 + n$}$

Calculating the Sum of the First Thirty-Two Terms

Now that we have the formula for the sum, we can calculate the sum of the first thirty-two terms, {S_{32}$}$. We can substitute n = 32 into the formula:

{S_{32} = \frac{32(32 + 1)}{2} \times 4 + 32 = 16 \times 1056 + 32 = 16896 + 32 = 16928$}$

Conclusion

In conclusion, we have successfully determined the thirty-first term of the sequence, {T_{31}$}$, and the sum of the first thirty-two terms, {S_{32}$}$. The thirty-first term is 125, and the sum of the first thirty-two terms is 16928.

References

Further Reading

Glossary

  • Arithmetic Sequence: A sequence of numbers in which the difference between consecutive terms is constant.
  • Geometric Sequence: A sequence of numbers in which the ratio between consecutive terms is constant.
  • Sequences and Series: A sequence of numbers in which the difference between consecutive terms is constant, or a sequence of numbers in which the ratio between consecutive terms is constant.
  • Term: A single element in a sequence.
  • Sum: The total value of a sequence.

Introduction

In our previous article, we explored the infinite number sequence ${1, 1, 1, 5, 1, 9, 1, 13, \ldots\$} and determined the thirty-first term of the sequence, {T_{31}$}$, and the sum of the first thirty-two terms, {S_{32}$}$. In this article, we will answer some frequently asked questions about the sequence and provide additional insights.

Q&A

Q: What is the pattern of the sequence?

A: The sequence can be broken down into two separate sequences: ${1, 1, 1, \ldots\$} and ${5, 9, 13, \ldots\$}. The first sequence consists of consecutive 1s, while the second sequence consists of consecutive odd numbers starting from 5.

Q: How do I calculate the nth term of the sequence?

A: To calculate the nth term of the sequence, you can use the formula {a_n = 4n + 1$}$ for the second sequence.

Q: How do I calculate the sum of the first n terms of the sequence?

A: To calculate the sum of the first n terms of the sequence, you can use the formula {S_n = \frac{n(n + 1)}{2} \times 4 + n$}$.

Q: What is the significance of the sequence?

A: The sequence has various applications in mathematics, such as in the study of arithmetic and geometric sequences, and in the calculation of sums and products of sequences.

Q: Can I use the sequence to solve real-world problems?

A: Yes, the sequence can be used to solve real-world problems, such as in finance, where the sequence can be used to calculate the future value of an investment, or in engineering, where the sequence can be used to calculate the sum of a series of loads.

Q: How do I determine the number of terms in the sequence?

A: To determine the number of terms in the sequence, you can use the formula {n = \frac{S_n}{\frac{n(n + 1)}{2} \times 4 + n}$}$.

Q: Can I use the sequence to calculate the sum of a series of numbers?

A: Yes, the sequence can be used to calculate the sum of a series of numbers. For example, if you have a series of numbers {a_1, a_2, a_3, \ldots, a_n$}$, you can use the sequence to calculate the sum {S_n = a_1 + a_2 + a_3 + \ldots + a_n$}$.

Conclusion

In conclusion, the infinite number sequence ${1, 1, 1, 5, 1, 9, 1, 13, \ldots\$} is a combination of two sequences, and it has various applications in mathematics and real-world problems. We hope that this Q&A article has provided you with a better understanding of the sequence and its significance.

References

Further Reading

Glossary

  • Arithmetic Sequence: A sequence of numbers in which the difference between consecutive terms is constant.
  • Geometric Sequence: A sequence of numbers in which the ratio between consecutive terms is constant.
  • Sequences and Series: A sequence of numbers in which the difference between consecutive terms is constant, or a sequence of numbers in which the ratio between consecutive terms is constant.
  • Term: A single element in a sequence.
  • Sum: The total value of a sequence.