Identify The 16th Term Of A Geometric Sequence Where $a_1=4$ And $a_8=-8,748$.A. − 172 , 186 , 884 -172,186,884 − 172 , 186 , 884 B. − 57 , 395 , 628 -57,395,628 − 57 , 395 , 628 C. 57 , 395 , 628 57,395,628 57 , 395 , 628 D. 172 , 186 , 884 172,186,884 172 , 186 , 884

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. In this article, we will explore how to identify the 16th term of a geometric sequence given the first term and the 8th term.

Understanding Geometric Sequences

A geometric sequence is defined by the formula:

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

where:

• $a_n$ is the nth term of the sequence
• $a_1$ is the first term of the sequence
• $r$ is the common ratio
• $n$ is the term number

Given Information

We are given the following information:

• $a_1 = 4$
• $a_8 = -8,748$

Finding the Common Ratio

To find the common ratio, we can use the formula:

$$r = \frac{a_n}{a_1}$$

Plugging in the values, we get:

$$r = \frac{-8,748}{4} = -2,187$$

Verifying the Common Ratio

To verify that the common ratio is indeed -2,187, we can use the formula:

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

Plugging in the values, we get:

$$a_8 = 4 \cdot (-2,187)^{(8-1)} = 4 \cdot (-2,187)^7 = -8,748$$

This confirms that the common ratio is indeed -2,187.

Finding the 16th Term

Now that we have the common ratio, we can find the 16th term using the formula:

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

Plugging in the values, we get:

$$a_{16} = 4 \cdot (-2,187)^{(16-1)} = 4 \cdot (-2,187)^{15}$$

Using a calculator, we get:

$$a_{16} = 4 \cdot (-2,187)^{15} = -57,395,628$$

Conclusion

In this article, we identified the 16th term of a geometric sequence given the first term and the 8th term. We found the common ratio to be -2,187 and used it to find the 16th term, which is -57,395,628.

Answer

The correct answer is:

• B. $-57,395,628$

Discussion

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

A: To find the common ratio, you can use the formula:

$$r = \frac{a_n}{a_1}$$

where:

• $r$ is the common ratio
• $a_n$ is the nth term of the sequence
• $a_1$ is the first term of the sequence

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

A: To find the nth term, you can use the formula:

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

where:

• $a_n$ is the nth term of the sequence
• $a_1$ is the first term of the sequence
• $r$ is the common ratio
• $n$ is the term number

Q: What is the formula for the sum of a geometric sequence?

A: The formula for the sum of a geometric sequence is:

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

where:

• $S_n$ is the sum of the first n terms
• $a_1$ is the first term of the sequence
• $r$ is the common ratio
• $n$ is the number of terms

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

A: To find the sum of an infinite geometric sequence, you can use the formula:

$$S = \frac{a_1}{1-r}$$

where:

• $S$ is the sum of the infinite sequence
• $a_1$ is the first term of the sequence
• $r$ is the common ratio

Q: What is the formula for the nth partial sum of a geometric sequence?

A: The formula for the nth partial sum of a geometric sequence is:

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

where:

• $S_n$ is the nth partial sum
• $a_1$ is the first term of the sequence
• $r$ is the common ratio
• $n$ is the number of terms

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

A: To find the common ratio, you can use the formula:

$$r = \frac{a_n}{a_m}$$

where:

• $r$ is the common ratio
• $a_n$ is the nth term of the sequence
• $a_m$ is the mth term of the sequence

Q: What is the relationship between the common ratio and the terms of a geometric sequence?

A: The common ratio is the ratio of any two consecutive terms in a geometric sequence. It is a fixed, non-zero number that is used to find each term in the sequence.

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

A: To determine if a sequence is geometric, you can check if the ratio of any two consecutive terms is constant. If it is, then the sequence is geometric.

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

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

• Compound interest
• Population growth
• Sales growth
• Electrical engineering
• Computer science

Q: How do I use geometric sequences in real-world problems?

A: To use geometric sequences in real-world problems, you can apply the formulas and concepts learned in this article to solve problems involving compound interest, population growth, sales growth, and other applications.

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 if the sequence is geometric before applying the formulas
• Not using the correct formula for the sum of a geometric sequence
• Not checking if the common ratio is valid before applying it
• Not using the correct formula for the nth partial sum of a geometric sequence

Q: How do I practice working with geometric sequences?

A: To practice working with geometric sequences, you can try the following:

• Work through examples and exercises in a textbook or online resource
• Practice finding the common ratio and the nth term of a geometric sequence
• Practice finding the sum of a geometric sequence
• Practice finding the nth partial sum of a geometric sequence
• Try solving real-world problems involving geometric sequences.