The N Th N^{\text{th}} N Th Term Rule For A Geometric Sequence Is 2 × 5 ( N − 1 ) 2 \times 5^{(n-1)} 2 × 5 ( N − 1 ) .Work Out The First Three Terms Of The Sequence.

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Introduction

A geometric sequence is a type of sequence where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio. The $n^{\text{th}}$ term rule for a geometric sequence is given by the formula $a_n = a_1 \times r^{(n-1)}$, where $a_n$ is the $n^{\text{th}}$ term, $a_1$ is the first term, $r$ is the common ratio, and $n$ is the term number. In this case, the $n^{\text{th}}$ term rule is $2 \times 5^{(n-1)}$. We are asked to work out the first three terms of the sequence.

The First Term

To find the first term, we substitute $n = 1$ into the $n^{\text{th}}$ term rule. This gives us:

$a_1 = 2 \times 5^{(1-1)}$ $a_1 = 2 \times 5^0$ $a_1 = 2 \times 1$ $a_1 = 2$

So, the first term of the sequence is 2.

The Second Term

To find the second term, we substitute $n = 2$ into the $n^{\text{th}}$ term rule. This gives us:

$a_2 = 2 \times 5^{(2-1)}$ $a_2 = 2 \times 5^1$ $a_2 = 2 \times 5$ $a_2 = 10$

So, the second term of the sequence is 10.

The Third Term

To find the third term, we substitute $n = 3$ into the $n^{\text{th}}$ term rule. This gives us:

$a_3 = 2 \times 5^{(3-1)}$ $a_3 = 2 \times 5^2$ $a_3 = 2 \times 25$ $a_3 = 50$

So, the third term of the sequence is 50.

Conclusion

In this article, we have worked out the first three terms of a geometric sequence using the $n^{\text{th}}$ term rule $2 \times 5^{(n-1)}$. We found that the first term is 2, the second term is 10, and the third term is 50. This demonstrates how to use the $n^{\text{th}}$ term rule to find the terms of a geometric sequence.

Example Use Case

The $n^{\text{th}}$ term rule can be used in a variety of situations, such as:

• Finding the value of a specific term in a geometric sequence
• Determining the common ratio of a geometric sequence
• Calculating the sum of the first n terms of a geometric sequence

For example, if we want to find the value of the 5th term of the sequence, we can substitute $n = 5$ into the $n^{\text{th}}$ term rule:

$a_5 = 2 \times 5^{(5-1)}$ $a_5 = 2 \times 5^4$ $a_5 = 2 \times 625$ $a_5 = 1250$

So, the 5th term of the sequence is 1250.

Common Ratio

The common ratio of a geometric sequence is the ratio of any term to its previous term. In this case, the common ratio is 5, since each term is obtained by multiplying the previous term by 5.

Sum of the First n Terms

The sum of the first n terms of a geometric sequence can be calculated using the formula:

$S_n = \frac{a_1(1-r^n)}{1-r}$

where $S_n$ is the sum of the first n terms, $a_1$ is the first term, $r$ is the common ratio, and $n$ is the number of terms.

For example, if we want to calculate the sum of the first 5 terms of the sequence, we can use the formula:

$S_5 = \frac{2(1-5^5)}{1-5}$ $S_5 = \frac{2(1-3125)}{-4}$ $S_5 = \frac{2(-3124)}{-4}$ $S_5 = 1562$

So, the sum of the first 5 terms of the sequence is 1562.

Conclusion

In this article, we have worked out the first three terms of a geometric sequence using the $n^{\text{th}}$ term rule $2 \times 5^{(n-1)}$. We have also discussed the common ratio and the sum of the first n terms of a geometric sequence. This demonstrates how to use the $n^{\text{th}}$ term rule to find the terms of a geometric sequence and how to calculate the sum of the first n terms.

Introduction

Geometric sequences are a fundamental concept in mathematics, and they have numerous applications in various fields such as finance, engineering, and computer science. In this article, we will answer some frequently asked questions about geometric sequences, including their definition, formula, and properties.

Q: What is a geometric sequence?

A: A geometric sequence is a type of sequence where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio.

Q: What is the formula for a geometric sequence?

A: The formula for a geometric sequence is given by:

$a_n = a_1 \times r^{(n-1)}$

where $a_n$ is the $n^{\text{th}}$ term, $a_1$ is the first term, $r$ is the common ratio, and $n$ is the term number.

Q: What is the common ratio in a geometric sequence?

A: The common ratio in a geometric sequence is the ratio of any term to its previous term. In other words, it is the factor by which each term is multiplied to get the next term.

Q: How do I find the common ratio of a geometric sequence?

A: To find the common ratio of a geometric sequence, you can use the following formula:

$r = \frac{a_n}{a_{n-1}}$

where $r$ is the common ratio, $a_n$ is the $n^{\text{th}}$ term, and $a_{n-1}$ is the $(n-1)^{\text{th}}$ term.

Q: What is the sum of the first n terms of a geometric sequence?

A: The sum of the first n terms of a geometric sequence can be calculated using the formula:

$S_n = \frac{a_1(1-r^n)}{1-r}$

where $S_n$ is the sum of the first n terms, $a_1$ is the first term, $r$ is the common ratio, and $n$ is the number of terms.

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

A: To calculate the sum of the first n terms of a geometric sequence, you can use the formula:

$S_n = \frac{a_1(1-r^n)}{1-r}$

where $S_n$ is the sum of the first n terms, $a_1$ is the first term, $r$ is the common ratio, and $n$ is the number of terms.

Q: What is the difference between a geometric sequence and an arithmetic sequence?

A: A geometric sequence is a type of sequence where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio. An arithmetic sequence, on the other hand, is a type of sequence where each term after the first is found by adding a fixed, non-zero number called the common difference.

Q: How do I determine if a sequence is geometric or arithmetic?

A: To determine if a sequence is geometric or arithmetic, you can look at the ratio of consecutive terms. If the ratio is constant, then the sequence is geometric. If the difference between consecutive terms is constant, then the sequence is arithmetic.

Q: What are some real-world applications of geometric sequences?

A: Geometric sequences have numerous real-world applications, including:

• Finance: Geometric sequences are used to calculate compound interest and investment returns.
• Engineering: Geometric sequences are used to model population growth and decay.
• Computer Science: Geometric sequences are used to model the growth of algorithms and data structures.

Conclusion

In this article, we have answered some frequently asked questions about geometric sequences, including their definition, formula, and properties. We have also discussed the common ratio, sum of the first n terms, and real-world applications of geometric sequences. This demonstrates the importance and relevance of geometric sequences in various fields.

Example Use Case

The following example illustrates how to use geometric sequences to calculate compound interest:

Suppose you invest $1000 at a 5% annual interest rate compounded annually. How much will you have after 5 years?

To calculate this, we can use the formula for the sum of the first n terms of a geometric sequence:

$S_n = \frac{a_1(1-r^n)}{1-r}$

where $S_n$ is the sum of the first n terms, $a_1$ is the first term ($1000$), $r$ is the common ratio ($1.05$), and $n$ is the number of terms ($5$).

Plugging in these values, we get:

$S_5 = \frac{1000(1-1.05^5)}{1-1.05}$ $S_5 = \frac{1000(1-1.2762815625)}{-0.05}$ $S_5 = \frac{1000(-0.2762815625)}{-0.05}$ $S_5 = 5527.62815625$

So, after 5 years, you will have approximately $5527.63.

Conclusion

In this article, we have demonstrated how to use geometric sequences to calculate compound interest and investment returns. This illustrates the importance and relevance of geometric sequences in finance and other fields.