Let's solve the problems step by step from the image.

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: $a_{15} = 44m + 5$

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 $a + 2d = 15$.

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.