What Is The 10th Term Of The Geometric Sequence A ( N ) = − 3 ( 2 ) N − 1 A(n) = -3(2)^{n-1} A ( N ) = − 3 ( 2 ) N − 1 ?

$$

Understanding Geometric Sequences

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 general formula for a geometric sequence is given by $a(n) = a_1 \cdot r^{n-1}$, where $a_1$ is the first term and $r$ is the common ratio. In this case, the given geometric sequence is $a(n) = -3(2)^{n-1}$.

Identifying the Common Ratio and First Term

To find the 10th term of the geometric sequence, we need to identify the common ratio and the first term. In this case, the common ratio is 2, and the first term is $-3(2)^{0} = -3$. The common ratio is the number by which each term is multiplied to get the next term, and in this case, it is 2.

Finding the 10th Term of the Geometric Sequence

To find the 10th term of the geometric sequence, we can use the formula $a(n) = a_1 \cdot r^{n-1}$. Plugging in the values, we get $a(10) = -3(2)^{10-1} = -3(2)^9$. To evaluate this expression, we need to calculate $2^9$.

Calculating $2^9$

To calculate $2^9$, we can use the fact that $2^9 = 2^8 \cdot 2^1$. Since $2^8 = 256$, we have $2^9 = 256 \cdot 2 = 512$.

Evaluating the 10th Term

Now that we have calculated $2^9$, we can evaluate the 10th term of the geometric sequence. Plugging in the value of $2^9$, we get $a(10) = -3(512) = -1536$.

Conclusion

In conclusion, the 10th term of the geometric sequence $a(n) = -3(2)^{n-1}$ is $-1536$. This can be verified by plugging in the value of $n = 10$ into the formula for the geometric sequence.

Example Use Case

The geometric sequence $a(n) = -3(2)^{n-1}$ can be used to model a variety of real-world situations, such as population growth or financial investments. For example, if we have a population that grows at a rate of 2% per year, we can use the geometric sequence to model the population growth over time.

Step-by-Step Solution

To find the 10th term of the geometric sequence, follow these steps:

1. Identify the common ratio and first term of the geometric sequence.
2. Use the formula $a(n) = a_1 \cdot r^{n-1}$ to find the nth term of the geometric sequence.
3. Plug in the values of $a_1$, $r$, and $n$ into the formula.
4. Evaluate the expression to find the nth term of the geometric sequence.

Common Ratio and First Term

The common ratio of the geometric sequence is 2, and the first term is $-3(2)^{0} = -3$.

Formula for the nth Term

The formula for the nth term of the geometric sequence is $a(n) = -3(2)^{n-1}$.

Evaluating the 10th Term

To evaluate the 10th term of the geometric sequence, we need to plug in the value of $n = 10$ into the formula. This gives us $a(10) = -3(2)^{10-1} = -3(2)^9$.

Calculating $2^9$

To calculate $2^9$, we can use the fact that $2^9 = 2^8 \cdot 2^1$. Since $2^8 = 256$, we have $2^9 = 256 \cdot 2 = 512$.

Evaluating the 10th Term

Now that we have calculated $2^9$, we can evaluate the 10th term of the geometric sequence. Plugging in the value of $2^9$, we get $a(10) = -3(512) = -1536$.

Conclusion

In conclusion, the 10th term of the geometric sequence $a(n) = -3(2)^{n-1}$ is $-1536$. This can be verified by plugging in the value of $n = 10$ into the formula for the geometric sequence.

Geometric Sequences in Real-World Applications

Geometric sequences have a wide range of applications in real-world situations, such as population growth, financial investments, and compound interest. For example, if we have a population that grows at a rate of 2% per year, we can use the geometric sequence to model the population growth over time.

Calculating Compound Interest

Compound interest is a type of interest that is calculated on both the principal amount and any accrued interest. The formula for compound interest is given by $A = P(1 + r)^n$, where $A$ is the amount of money accumulated after $n$ years, $P$ is the principal amount, $r$ is the annual interest rate, and $n$ is the number of years.

Example of Compound Interest

Suppose we have a principal amount of $1000 and an annual interest rate of 5%. We want to calculate the amount of money accumulated after 5 years. Using the formula for compound interest, we get $A = 1000(1 + 0.05)^5 = 1000(1.05)^5$.

Calculating $(1.05)^5$

To calculate $(1.05)^5$, we can use the fact that $(1.05)^5 = (1.05)^4 \cdot (1.05)^1$. Since $(1.05)^4 = 1.21550625$, we have $(1.05)^5 = 1.21550625 \cdot 1.05 = 1.2815065625$.

Evaluating the Amount of Money Accumulated

Now that we have calculated $(1.05)^5$, we can evaluate the amount of money accumulated after 5 years. Plugging in the value of $(1.05)^5$, we get $A = 1000(1.2815065625) = 1281.5065625$.

Conclusion

In conclusion, the amount of money accumulated after 5 years is $1281.5065625$. This can be verified by plugging in the values of $P$, $r$, and $n$ into the formula for compound interest.

Geometric Sequences and Compound Interest

Geometric sequences and compound interest are closely related concepts. The formula for compound interest is given by $A = P(1 + r)^n$, which is similar to the formula for the nth term of a geometric sequence, $a(n) = a_1 \cdot r^{n-1}$. In fact, the formula for compound interest can be rewritten as $A = P(1 + r)^n = P(1 + r)^{n-1} \cdot (1 + r) = a_1 \cdot r^{n-1} \cdot (1 + r)$.

Conclusion

In conclusion, geometric sequences and compound interest are closely related concepts. The formula for compound interest can be rewritten as a geometric sequence, and the two concepts can be used to model a variety of real-world situations, such as population growth and financial investments.

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 $a(n) = a_1 \cdot r^{n-1}$, where $a_1$ is the first term and $r$ is the common ratio.

Q: How do I find the nth term of a geometric sequence?

A: To find the nth term of a geometric sequence, you can use the formula $a(n) = a_1 \cdot r^{n-1}$. Simply plug in the values of $a_1$, $r$, and $n$ into the formula.

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

A: The common ratio in a geometric sequence is the number by which each term is multiplied to get the next term. It is denoted by the variable $r$.

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

A: To find the common ratio of a geometric sequence, you can divide any term by the previous term. For example, if you have the terms $a_1$ and $a_2$, you can find the common ratio by dividing $a_2$ by $a_1$.

Q: What is the first term of a geometric sequence?

A: The first term of a geometric sequence is the initial term of the sequence. It is denoted by the variable $a_1$.

Q: How do I find the first term of a geometric sequence?

A: To find the first term of a geometric sequence, you can use the formula $a_1 = \frac{a(n)}{r^{n-1}}$. Simply plug in the values of $a(n)$, $r$, and $n$ into the formula.

Q: What is the sum of a geometric sequence?

A: The sum of a geometric sequence is the sum of all the terms in the sequence. It is denoted by the variable $S_n$.

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

A: To find the sum of a geometric sequence, you can use the formula $S_n = \frac{a_1(1 - r^n)}{1 - r}$. Simply plug in the values of $a_1$, $r$, and $n$ into the formula.

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 number called the common difference.

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

A: To determine whether a sequence is geometric or arithmetic, you can look at the relationship between the terms. If the terms are related by multiplication, the sequence is geometric. If the terms are related by addition, the sequence is arithmetic.

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

A: Geometric sequences have a wide range of applications in real-world situations, such as population growth, financial investments, and compound interest.

Q: How do I use geometric sequences to model population growth?

A: To use geometric sequences to model population growth, you can use the formula $a(n) = a_1 \cdot r^{n-1}$, where $a_1$ is the initial population and $r$ is the growth rate.

Q: How do I use geometric sequences to model financial investments?

A: To use geometric sequences to model financial investments, you can use the formula $a(n) = a_1 \cdot r^{n-1}$, where $a_1$ is the initial investment and $r$ is the interest rate.

Q: What are some common mistakes to avoid when working with geometric sequences?

A: Some common mistakes to avoid when working with geometric sequences include:

• Not checking for the existence of the common ratio
• Not checking for the existence of the first term
• Not using the correct formula for the nth term
• Not using the correct formula for the sum of the sequence

Q: How do I troubleshoot common mistakes when working with geometric sequences?

A: To troubleshoot common mistakes when working with geometric sequences, you can:

• Check the existence of the common ratio and the first term
• Use the correct formula for the nth term and the sum of the sequence
• Verify the calculations and the results
• Seek help from a teacher or a tutor if needed

Q: What are some advanced topics related to geometric sequences?

A: Some advanced topics related to geometric sequences include:

• Inequalities involving geometric sequences
• Equations involving geometric sequences
• Systems of equations involving geometric sequences
• Geometric sequences with complex numbers

Q: How do I learn more about geometric sequences?

A: To learn more about geometric sequences, you can:

• Read books and online resources
• Watch video lectures and tutorials
• Practice problems and exercises
• Join online communities and forums
• Seek help from a teacher or a tutor