The Nth Term Of The Sequence ${ 9, 9^4, 9^7, 9^{10}, \ldots\$} Is Given By ${ 9^{3n-2}\$} . (Simplify Your Answer If Needed.)Find The 100th Term Of The Sequence [$188 + 5 \cdot 2^{29}, 188 + 6 \cdot 2^{29}, 188 + 7 \cdot 2^{29},
Introduction
In mathematics, sequences are an essential concept that helps us understand patterns and relationships between numbers. A sequence is a list of numbers in a specific order, and each term in the sequence is determined by a specific rule or formula. In this article, we will explore the nth term of a given sequence and find the 100th term of the sequence.
The Sequence
The given sequence is ${9, 9^4, 9^7, 9^{10}, \ldots\$}. To find the nth term of this sequence, we can use the formula ${9^{3n-2}\$}. This formula indicates that each term in the sequence is obtained by raising 9 to the power of 3n-2, where n is the position of the term in the sequence.
Understanding the Formula
The formula $9^{3n-2}\$} can be broken down into two parts$}$ and ${9^{-2}\$}. The first part, ${9^{3n}\$}, represents the power of 9 raised to the power of 3n. The second part, ${9^{-2}\$}, represents the reciprocal of 9 squared.
Simplifying the Formula
To simplify the formula, we can rewrite it as ${9^{3n} \cdot 9^{-2}\$}. Using the properties of exponents, we can combine the two terms as ${9^{3n-2}\$}.
Finding the 100th Term
Now that we have the formula for the nth term, we can find the 100th term of the sequence. To do this, we substitute n=100 into the formula:
${$9^{3(100)-2}$ = 9^{298}$.
Calculating the Value
To calculate the value of [9^{298}\$}, we can use a calculator or a computer program. However, for the sake of this article, we will leave the calculation as is.
The 100th Term of the Sequence
The 100th term of the sequence is ${9^{298}\$}. This value represents the 100th term in the sequence ${9, 9^4, 9^7, 9^{10}, \ldots\$}.
Discussion
The sequence ${9, 9^4, 9^7, 9^{10}, \ldots\$} is an example of a geometric sequence, where each term is obtained by multiplying the previous term by a fixed constant. In this case, the constant is 9. The formula ${9^{3n-2}\$} provides a general rule for finding any term in the sequence.
Conclusion
In conclusion, we have explored the nth term of a given sequence and found the 100th term of the sequence. The formula ${9^{3n-2}\$} provides a general rule for finding any term in the sequence, and we have used this formula to calculate the 100th term.
**The Sequence ${
The sequence [188 + 5 \cdot 2^{29}, 188 + 6 \cdot 2^{29}, 188 + 7 \cdot 2^{29}, \ldots\$} is another example of a sequence that can be analyzed using the concepts of sequences and series. However, this sequence is not a geometric sequence, and its terms are not obtained by multiplying the previous term by a fixed constant.
Understanding the Sequence
To understand this sequence, we need to analyze its terms and look for patterns or relationships between them. The terms of the sequence are given by ${188 + n \cdot 2^{29}\$}, where n is the position of the term in the sequence.
Finding the nth Term
To find the nth term of this sequence, we can use the formula ${188 + n \cdot 2^{29}\$}. This formula indicates that each term in the sequence is obtained by adding n times ${2^{29}\$} to 188.
Calculating the Value
To calculate the value of the nth term, we can substitute n into the formula:
${$188 + n \cdot 2^{29}$.
The nth Term of the Sequence
The nth term of the sequence is [188 + n \cdot 2^{29}\$}. This value represents the nth term in the sequence ${188 + 5 \cdot 2^{29}, 188 + 6 \cdot 2^{29}, 188 + 7 \cdot 2^{29}, \ldots\$}.
Discussion
The sequence ${188 + 5 \cdot 2^{29}, 188 + 6 \cdot 2^{29}, 188 + 7 \cdot 2^{29}, \ldots\$} is an example of a sequence that can be analyzed using the concepts of sequences and series. The formula ${188 + n \cdot 2^{29}\$} provides a general rule for finding any term in the sequence.
Conclusion
Q: What is the nth term of a sequence?
A: The nth term of a sequence is the term that appears in the sequence at the nth position. It is the value of the term that is obtained by applying the formula or rule of the sequence to the nth position.
Q: How do I find the nth term of a sequence?
A: To find the nth term of a sequence, you need to apply the formula or rule of the sequence to the nth position. This may involve substituting the value of n into the formula or using a specific rule or pattern to determine the value of the nth term.
Q: What is the formula for the nth term of the sequence ${9, 9^4, 9^7, 9^{10}, \ldots\$}?
A: The formula for the nth term of the sequence ${9, 9^4, 9^7, 9^{10}, \ldots\$} is ${9^{3n-2}\$}.
Q: How do I calculate the value of the nth term of the sequence ${9, 9^4, 9^7, 9^{10}, \ldots\$}?
A: To calculate the value of the nth term of the sequence ${9, 9^4, 9^7, 9^{10}, \ldots\$}, you need to substitute the value of n into the formula ${9^{3n-2}\$} and calculate the result.
Q: What is the 100th term of the sequence ${9, 9^4, 9^7, 9^{10}, \ldots\$}?
A: The 100th term of the sequence ${9, 9^4, 9^7, 9^{10}, \ldots\$} is ${9^{298}\$}.
Q: How do I find the nth term of the sequence ${188 + 5 \cdot 2^{29}, 188 + 6 \cdot 2^{29}, 188 + 7 \cdot 2^{29}, \ldots\$}?
A: To find the nth term of the sequence ${188 + 5 \cdot 2^{29}, 188 + 6 \cdot 2^{29}, 188 + 7 \cdot 2^{29}, \ldots\$}, you need to apply the formula ${188 + n \cdot 2^{29}\$} to the nth position.
Q: What is the formula for the nth term of the sequence ${188 + 5 \cdot 2^{29}, 188 + 6 \cdot 2^{29}, 188 + 7 \cdot 2^{29}, \ldots\$}?
A: The formula for the nth term of the sequence ${188 + 5 \cdot 2^{29}, 188 + 6 \cdot 2^{29}, 188 + 7 \cdot 2^{29}, \ldots\$} is ${188 + n \cdot 2^{29}\$}.
Q: How do I calculate the value of the nth term of the sequence ${188 + 5 \cdot 2^{29}, 188 + 6 \cdot 2^{29}, 188 + 7 \cdot 2^{29}, \ldots\$}?
A: To calculate the value of the nth term of the sequence ${188 + 5 \cdot 2^{29}, 188 + 6 \cdot 2^{29}, 188 + 7 \cdot 2^{29}, \ldots\$}, you need to substitute the value of n into the formula ${188 + n \cdot 2^{29}\$} and calculate the result.
Q: What is the nth term of the sequence ${9, 9^4, 9^7, 9^{10}, \ldots\$} and the sequence ${188 + 5 \cdot 2^{29}, 188 + 6 \cdot 2^{29}, 188 + 7 \cdot 2^{29}, \ldots\$}?
A: The nth term of the sequence ${9, 9^4, 9^7, 9^{10}, \ldots\$} is ${9^{3n-2}\$} and the nth term of the sequence ${188 + 5 \cdot 2^{29}, 188 + 6 \cdot 2^{29}, 188 + 7 \cdot 2^{29}, \ldots\$} is ${188 + n \cdot 2^{29}\$}.
Conclusion
In conclusion, we have provided answers to some common questions about the nth term of a sequence. We have also provided formulas and examples for finding the nth term of two different sequences. We hope that this Q&A article has been helpful in understanding the concept of the nth term of a sequence.