https://file.aiquickdraw.com/mathbot/file/1725581209872vfh2wseu.jpg

Let's address each question in the image:

1. **What is the arithmetic mean between 15 and 63?**

   The arithmetic mean between two numbers \( a \) and \( b \) is calculated as:
   \[
   \text{Mean} = \frac{a + b}{2}
   \]
   So, between 15 and 63, the mean is:
   \[
   \frac{15 + 63}{2} = \frac{78}{2} = 39
   \]
   **Answer:** The arithmetic mean is **39**.

2. **Insert four arithmetic means between 35 and 95.**

   When inserting \( n \) arithmetic means between two numbers, the sequence is divided into \( n+1 \) equal intervals. In this case, we want to insert 4 arithmetic means, so the sequence will have 6 terms.

   Let the common difference be \( d \). The sequence will be: 35, \( a_2 \), \( a_3 \), \( a_4 \), \( a_5 \), 95.

   The general formula for the \( n \)-th term in an arithmetic sequence is:
   \[
   a_n = a_1 + (n-1)d
   \]
   For the 6th term:
   \[
   95 = 35 + 5d
   \]
   Solving for \( d \):
   \[
   95 - 35 = 5d \quad \Rightarrow \quad 60 = 5d \quad \Rightarrow \quad d = 12
   \]
   So, the sequence will be:
   \[
   35, 47, 59, 71, 83, 95
   \]
   **Answer:** The four arithmetic means are **47, 59, 71, 83**.

3. **Find the missing terms of the arithmetic sequence: __, 15, __, __, 63.**

   Let's denote the missing terms as \( a_1 \), \( a_3 \), and \( a_4 \). Since this is an arithmetic sequence, the difference between consecutive terms is constant. Let the common difference be \( d \).

   The sequence looks like: \( a_1, 15, a_3, a_4, 63 \).

   Using the formula for the \( n \)-th term:
   \[
   a_5 = a_1 + 4d = 63
   \]
   Also,
   \[
   a_2 = a_1 + d = 15
   \]
   From \( a_1 + d = 15 \), we get:
   \[
   d = 15 - a_1
   \]
   Substituting into the equation \( a_1 + 4d = 63 \):
   \[
   a_1 + 4(15 - a_1) = 63
   \]
   Expanding and solving for \( a_1 \):
   \[
   a_1 + 60 - 4a_1 = 63
   \]
   \[
   -3a_1 + 60 = 63 \quad \Rightarrow \quad -3a_1 = 3 \quad \Rightarrow \quad a_1 = -1
   \]
   So, \( d = 15 - (-1) = 16 \).

   The full sequence is:
   \[
   -1, 15, 31, 47, 63
   \]
   **Answer:** The missing terms are **-1, 31, 47**.

Would you like further details on these solutions or have any questions?

### Related Questions:
1. What is the formula for the sum of the first \( n \) terms of an arithmetic sequence?
2. How do you calculate the common difference if the first and last terms of an arithmetic sequence are known?
3. Can you find the 10th term of the sequence that starts with -1 and has a common difference of 16?
4. How do you calculate the arithmetic mean of more than two numbers?
5. How would you calculate the nth term of an arithmetic sequence given its first term and common difference?

### Tip:
Always remember that in an arithmetic sequence, the difference between consecutive terms remains constant throughout the sequence.

1. What is the arithmetic mean between 15 and 63?
2. Insert four arithmetic means between 35 and 95.
3. Find the missing terms of the arithmetic sequence __, 15, __, __, 63.

Learn how to calculate the arithmetic mean between 15 and 63, insert four arithmetic means between 35 and 95, and find the missing terms of an arithmetic sequence. This tutorial covers key arithmetic concepts and formulas. Suitable for students in Grades 7-9.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation