Find The 6th Term Of This Geometric Sequence.$ \begin{array}{c} -1, 4, -16, 64, \ldots \ a_6 = [?] \ \text{Hint A_n = A_1 \cdot R^{(n-1)} \end{array} }$

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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 formula for the nth term of a geometric sequence is given by:

a_n = a_1 * r^(n-1)

where a_n is the nth term, a_1 is the first term, r is the common ratio, and n is the term number.

Identifying the First Term and Common Ratio

In the given geometric sequence, the first term is -1 and the common ratio can be found by dividing the second term by the first term:

r = a_2 / a_1 = 4 / -1 = -4

Finding the 6th Term

Now that we have the first term and the common ratio, we can use the formula to find the 6th term:

a_6 = a_1 * r^(n-1) a_6 = -1 * (-4)^(6-1) a_6 = -1 * (-4)^5 a_6 = -1 * (-1024) a_6 = 1024

Therefore, the 6th term of the geometric sequence is 1024.

Real-World Applications of Geometric Sequences

Geometric sequences have many real-world applications, including:

  • Finance: Geometric sequences can be used to model the growth of investments, such as compound interest.
  • Biology: Geometric sequences can be used to model the growth of populations, such as the spread of a disease.
  • Physics: Geometric sequences can be used to model the motion of objects, such as the trajectory of a projectile.

Conclusion

In conclusion, finding the 6th term of a geometric sequence involves identifying the first term and the common ratio, and then using the formula to calculate the nth term. Geometric sequences have many real-world applications and are an important concept in mathematics.

Example Problems

  1. Find the 8th term of the geometric sequence: 2, 6, 18, 54, ...
  2. Find the 10th term of the geometric sequence: 3, 9, 27, 81, ...
  3. Find the 12th term of the geometric sequence: 4, 16, 64, 256, ...

Solutions

  1. a_8 = a_1 * r^(n-1) a_8 = 2 * 3^(8-1) a_8 = 2 * 3^7 a_8 = 2 * 2187 a_8 = 4374
  2. a_10 = a_1 * r^(n-1) a_10 = 3 * 3^(10-1) a_10 = 3 * 3^9 a_10 = 3 * 19683 a_10 = 59049
  3. a_12 = a_1 * r^(n-1) a_12 = 4 * 4^(12-1) a_12 = 4 * 4^11 a_12 = 4 * 4194304 a_12 = 16777216

Practice Problems

  1. Find the 5th term of the geometric sequence: 2, 6, 18, 54, ...
  2. Find the 7th term of the geometric sequence: 3, 9, 27, 81, ...
  3. Find the 9th term of the geometric sequence: 4, 16, 64, 256, ...

Answers

  1. a_5 = a_1 * r^(n-1) a_5 = 2 * 3^(5-1) a_5 = 2 * 3^4 a_5 = 2 * 81 a_5 = 162
  2. a_7 = a_1 * r^(n-1) a_7 = 3 * 3^(7-1) a_7 = 3 * 3^6 a_7 = 3 * 729 a_7 = 2187
  3. a_9 = a_1 * r^(n-1) a_9 = 4 * 4^(9-1) a_9 = 4 * 4^8 a_9 = 4 * 65536 a_9 = 262144

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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, divide the second term by the first term. For example, if the first term is 2 and the second term is 6, the common ratio is 6/2 = 3.

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

A: To find the nth term, use the formula: a_n = a_1 * r^(n-1), where a_n is the nth term, a_1 is the first term, r is the common ratio, and 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 = 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 find the sum of an infinite geometric sequence?

A: To find the sum of an infinite geometric sequence, use the formula: S = a_1 / (1 - r), where S is the sum, a_1 is the first term, and r is the common ratio.

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

A: A geometric sequence is a sequence where each term is found by multiplying the previous term by a fixed number, while an arithmetic sequence is a sequence where each term is found by adding a fixed number to the previous term.

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

A: To determine if a sequence is geometric or arithmetic, look for a pattern in the differences or ratios between consecutive terms. If the ratios are constant, the sequence is geometric. If the differences are constant, the sequence is arithmetic.

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

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

  • Finance: Geometric sequences can be used to model the growth of investments, such as compound interest.
  • Biology: Geometric sequences can be used to model the growth of populations, such as the spread of a disease.
  • Physics: Geometric sequences can be used to model the motion of objects, such as the trajectory of a projectile.

Q: How do I find the 6th term of a geometric sequence with a first term of 2 and a common ratio of 3?

A: To find the 6th term, use the formula: a_n = a_1 * r^(n-1). Plugging in the values, we get: a_6 = 2 * 3^(6-1) = 2 * 3^5 = 2 * 243 = 486.

Q: How do I find the sum of the first 5 terms of a geometric sequence with a first term of 3 and a common ratio of 2?

A: To find the sum, use the formula: S_n = a_1 * (1 - r^n) / (1 - r). Plugging in the values, we get: S_5 = 3 * (1 - 2^5) / (1 - 2) = 3 * (1 - 32) / (-1) = 3 * (-31) / (-1) = 93.

Q: How do I find the sum of an infinite geometric sequence with a first term of 4 and a common ratio of 1/2?

A: To find the sum, use the formula: S = a_1 / (1 - r). Plugging in the values, we get: S = 4 / (1 - 1/2) = 4 / (1/2) = 4 * 2 = 8.

Q: What is the formula for the nth term of a geometric sequence with a first term of 5 and a common ratio of 4?

A: The formula for the nth term is: a_n = a_1 * r^(n-1). Plugging in the values, we get: a_n = 5 * 4^(n-1).

Q: How do I find the sum of the first 3 terms of a geometric sequence with a first term of 2 and a common ratio of 3?

A: To find the sum, use the formula: S_n = a_1 * (1 - r^n) / (1 - r). Plugging in the values, we get: S_3 = 2 * (1 - 3^3) / (1 - 3) = 2 * (1 - 27) / (-2) = 2 * (-26) / (-2) = 26.