https://file.aiquickdraw.com/mathbot/file/1726016854626x14ho7od.jpg

sequence:%201,%2054,%2064,%20...%2012th%20term

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### Part II: Find the indicated term of each item

1. **Arithmetic sequence: 10, 13, 16, ...**
   - Common difference \( d = 13 - 10 = 3 \)
   - The \( n \)-th term of an arithmetic sequence is given by:
     \[
     a_n = a_1 + (n - 1) \cdot d
     \]
     For the 10th term:
     \[
     a_{10} = 10 + (10 - 1) \cdot 3 = 10 + 27 = 37
     \]
     **Answer: \( a_{10} = 37 \)**

2. **Sequence: \( 1, \frac{5}{4}, \frac{6}{4}, \dots \)**
   - This appears to be a fractional arithmetic sequence where the common difference \( d = \frac{5}{4} - 1 = \frac{1}{4} \).
   - Using the arithmetic sequence formula, for the 12th term:
     \[
     a_{12} = 1 + (12 - 1) \cdot \frac{1}{4} = 1 + \frac{11}{4} = \frac{4}{4} + \frac{11}{4} = \frac{15}{4}
     \]
     **Answer: \( a_{12} = \frac{15}{4} \)**

3. **Arithmetic sequence: \( 3x, 8x, 13x, \dots \)**
   - Common difference \( d = 8x - 3x = 5x \).
   - For the 13th term:
     \[
     a_{13} = 3x + (13 - 1) \cdot 5x = 3x + 60x = 63x
     \]
     **Answer: \( a_{13} = 63x \)**

4. **Sequence: \( 2m + 5, 5m + 5, 8m + 5, \dots \)**
   - This is an arithmetic sequence where the common difference \( d = (5m + 5) - (2m + 5) = 3m \).
   - For the 15th term:
     \[
     a_{15} = (2m + 5) + (15 - 1) \cdot 3m = (2m + 5) + 42m = 44m + 5
     \]
     **Answer: \( a_{15} = 44m + 5 \)**

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### Part III: Answer each problem

1. **Insert four arithmetic means between 3 and 83.**
   - Total terms = 6 (including 3 and 83).
   - Common difference \( d \) is calculated as:
     \[
     d = \frac{83 - 3}{6 - 1} = \frac{80}{5} = 16
     \]
     The four arithmetic means are:
     \[
     3 + 16 = 19,\quad 19 + 16 = 35,\quad 35 + 16 = 51,\quad 51 + 16 = 67
     \]
     **Answer: 19, 35, 51, 67**

2. **Insert three arithmetic means between -44 and -12.**
   - Total terms = 5 (including -44 and -12).
   - Common difference \( d \) is:
     \[
     d = \frac{-12 - (-44)}{5 - 1} = \frac{32}{4} = 8
     \]
     The three arithmetic means are:
     \[
     -44 + 8 = -36,\quad -36 + 8 = -28,\quad -28 + 8 = -20
     \]
     **Answer: -36, -28, -20**

3. **The arithmetic mean of the second and fourth term of an arithmetic sequence is 15.**
   - Let the first term be \( a \) and the common difference be \( d \).
   - Second term: \( a + d \), fourth term: \( a + 3d \).
   - The arithmetic mean is:
     \[
     \frac{(a + d) + (a + 3d)}{2} = 15
     \]
     Simplifying:
     \[
     \frac{2a + 4d}{2} = 15 \implies a + 2d = 15
     \]
     Therefore, the relationship between \( a \) and \( d \) is \( a + 2d = 15 \).

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Do you need any more details or have any questions? Here are 5 related questions to explore:

1. What is the general formula for finding the \( n \)-th term in an arithmetic sequence?
2. How can you determine if a sequence is arithmetic or geometric?
3. What is the difference between an arithmetic mean and a geometric mean?
4. Can you find the sum of the first \( n \) terms of an arithmetic sequence?
5. How can you derive the common difference if only partial terms of a sequence are given?

**Tip:** Always check if a sequence is arithmetic by verifying the common difference between terms.

Find the indicated terms and solve arithmetic sequence problems, including finding the 10th, 12th, 13th, and 15th terms of different sequences and inserting arithmetic means between given numbers.

Learn how to solve arithmetic sequence problems, including finding the 10th, 12th, 13th, and 15th terms. Understand the process of inserting arithmetic means between given numbers, using arithmetic sequence formulas. Suitable for students in Grades 9-11.

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