https://file.aiquickdraw.com/mathbot/file/1723121190493xl46t531.jpg

Can you solve this by Arithmetic sequence

Sure, let's analyze the sequences in the provided image to determine if they follow an arithmetic sequence and solve for the common difference \(d\). Then, we will find the next three terms for each sequence.

### Sequence 1: 14, 25, 36, 47, ...

First, we check if the sequence is arithmetic by calculating the differences between consecutive terms:
- \(25 - 14 = 11\)
- \(36 - 25 = 11\)
- \(47 - 36 = 11\)

The common difference \(d\) is 11.

The next three terms:
\[ 47 + 11 = 58 \]
\[ 58 + 11 = 69 \]
\[ 69 + 11 = 80 \]

Thus, the sequence continues as:
\[ 14, 25, 36, 47, 58, 69, 80 \]

### Sequence 2: -34, -29, -24, -19, ...

Check the differences:
- \(-29 - (-34) = 5\)
- \(-24 - (-29) = 5\)
- \(-19 - (-24) = 5\)

The common difference \(d\) is 5.

The next three terms:
\[ -19 + 5 = -14 \]
\[ -14 + 5 = -9 \]
\[ -9 + 5 = -4 \]

Thus, the sequence continues as:
\[ -34, -29, -24, -19, -14, -9, -4 \]

### Sequence 3: 27, 17, 7, -3, ...

Check the differences:
- \(17 - 27 = -10\)
- \(7 - 17 = -10\)
- \(-3 - 7 = -10\)

The common difference \(d\) is -10.

The next three terms:
\[ -3 - 10 = -13 \]
\[ -13 - 10 = -23 \]
\[ -23 - 10 = -33 \]

Thus, the sequence continues as:
\[ 27, 17, 7, -3, -13, -23, -33 \]

### Sequence 4: \(\frac{1}{2}, \frac{1}{5}, \frac{1}{8}, \frac{1}{11}, \)

Check the differences:
- \(\frac{1}{5} - \frac{1}{2} = \frac{2 - 5}{10} = -\frac{3}{10}\)
- \(\frac{1}{8} - \frac{1}{5} = \frac{5 - 8}{40} = -\frac{3}{40}\)
- \(\frac{1}{11} - \frac{1}{8} = \frac{8 - 11}{88} = -\frac{3}{88}\)

Here, the common difference is not consistent, so it is not an arithmetic sequence.

### Sequence 5: \( x, 2, x + 4 \)

Check the differences:
- \( 2 - x \)
- \( (x + 4) - 2 = x + 2 \)

To find \(x\), we set the differences equal since they must be the same for an arithmetic sequence:
\[ 2 - x = x + 2 \]
\[ 2 - x = x + 2 \]
\[ 2 = 2x + 2 \]
\[ 0 = 2x \]
\[ x = 0 \]

So, the sequence becomes:
\[ 0, 2, 4 \]

The next three terms:
\[ 4 + 2 = 6 \]
\[ 6 + 2 = 8 \]
\[ 8 + 2 = 10 \]

Thus, the sequence continues as:
\[ 0, 2, 4, 6, 8, 10 \]

Would you like any further details or have any questions?

Here are 8 related questions to expand on the previous information:
1. Can you identify other types of sequences apart from arithmetic ones?
2. How can you determine if a sequence is geometric instead of arithmetic?
3. What are the formulas for finding the \(n\)th term of an arithmetic sequence?
4. How would the approach differ for sequences with fractional or decimal common differences?
5. Can you provide an example of a quadratic sequence and its general formula?
6. How do arithmetic sequences relate to real-world applications?
7. What is the sum of the first \(n\) terms in an arithmetic sequence?
8. How can arithmetic sequences be used to solve algebraic problems?

**Tip:** For verifying if a sequence is arithmetic, always check if the difference between consecutive terms remains constant throughout the sequence.

Explore how to solve arithmetic sequences by analyzing given sequences and calculating the next terms. This detailed guide covers the arithmetic sequence nth term formula and provides step-by-step solutions for various sequence examples. Suitable for high school students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation