Geometric Sequence And Series Problems: Solutions And Explanations

Oct 19, 2025 by Dimemap Team 67 views

Hey guys! Let's dive into some cool math problems involving geometric sequences and series. We'll break down each question step by step so you can totally nail these types of problems. Get ready to sharpen those pencils!

1. Finding the 9th Term of a Geometric Sequence

Okay, so the first question asks us to find the 9th term of the geometric sequence: 256, 384, 576, ...

Geometric sequences are sequences where each term is multiplied by a constant to get the next term. This constant is called the common ratio. To find any term in a geometric sequence, we use the formula:

Uₙ = a * r^(n-1)

Where:

Uₙ is the nth term of the sequence.
a is the first term of the sequence.
r is the common ratio.
n is the term number we want to find.

Step-by-Step Solution

Identify the first term (a): In our sequence, the first term a is 256.
Calculate the common ratio (r): To find r, we divide any term by its preceding term. Let's divide the second term (384) by the first term (256):
```
r = 384 / 256 = 1.5
```
So, the common ratio r is 1.5.
Find the 9th term (U₉): Now we use the formula to find the 9th term:
```
U₉ = 256 * (1.5)^(9-1)
U₉ = 256 * (1.5)^8
U₉ = 256 * 25.62890625
U₉ = 6561
```
Therefore, the 9th term of the geometric sequence is 6561.

So, the correct answer is A. 6.561. Understanding geometric sequences is super helpful because they pop up everywhere, from calculating compound interest to modeling population growth. Remembering the formula and knowing how to find the common ratio will make these problems a breeze! You got this!

2. Determining the 13th Term of a Sequence

Alright, let's tackle another sequence problem. This time, we need to find the 13th term of the sequence: 1/6, 1/3, 1/2, 2/3, ...

When we look at this sequence, the difference between consecutive terms appears to be constant. This suggests that it might be an arithmetic sequence. An arithmetic sequence is a sequence where the difference between consecutive terms is constant. This constant difference is called the common difference.

To find any term in an arithmetic sequence, we use the formula:

Uₙ = a + (n - 1) * d

Where:

Uₙ is the nth term of the sequence.
a is the first term of the sequence.
d is the common difference.
n is the term number we want to find.

Step-by-Step Solution

Identify the first term (a): The first term a is 1/6.
Calculate the common difference (d): To find d, we subtract any term from its succeeding term. Let's subtract the first term (1/6) from the second term (1/3):
```
d = 1/3 - 1/6 = 2/6 - 1/6 = 1/6
```
The common difference d is 1/6.
Find the 13th term (U₁₃): Using the formula, we can find the 13th term:
```
U₁₃ = 1/6 + (13 - 1) * (1/6)
U₁₃ = 1/6 + 12 * (1/6)
U₁₃ = 1/6 + 12/6
U₁₃ = 13/6
```
So, the 13th term of the arithmetic sequence is 13/6.

Therefore, the 13th term of the sequence is 13/6. Recognizing arithmetic sequences and applying the correct formula is crucial for solving these types of problems. Keep practicing, and you'll become a sequence-solving pro!

3. Solving for a Term in a Geometric Series

Now, let's dive into a problem involving a geometric series. We're given that in a geometric series, the 2nd term (U₂) is 6 and the 4th term (U₄) is 9. The goal is to find the 6th term (U₆).

In a geometric series, each term is obtained by multiplying the previous term by a constant ratio. We can express the terms as follows:

U₂ = a * r
U₄ = a * r³
U₆ = a * r⁵

Where:

a is the first term.
r is the common ratio.

Step-by-Step Solution

Set up the equations:

We know:
```
U₂ = a * r = 6
U₄ = a * r³ = 9
```
Solve for the common ratio (r):

Divide the second equation by the first equation:
```
(a * r³) / (a * r) = 9 / 6
r² = 3/2
r = √(3/2)
```
Solve for the first term (a):

Substitute the value of r into the first equation:
```
a * √(3/2) = 6
a = 6 / √(3/2)
a = 6 * √(2/3)
```

Find the 6th term (U₆):

Now, use the formula for the 6th term:

U₆ = a * r⁵
U₆ = (6 * √(2/3)) * (√(3/2))⁵
U₆ = 6 * (√(2/3)) * (√(3/2)) * (√(3/2)) * (√(3/2)) * (√(3/2)) * (√(3/2))
U₆ = 6 * (3/2)²
U₆ = 6 * (9/4)
U₆ = 54/4
U₆ = 27/2
U₆ = 13.5

So, 16U₆ = 16 * 13.5 = 216

Therefore, the 6th term (U₆) of the geometric series is 13.5. Being able to manipulate equations and solve for unknowns is a fundamental skill in math. Keep practicing these techniques, and you'll become a master problem-solver!

Wrapping things up, remember that practice makes perfect. Keep working on these types of problems, and you'll become more confident and skilled in no time. You've got this!