In the five-sided star shown, the letters A, B, C, D and E are replaced by the numbers 3,5,6,7, and 9, although not necessarily in this order. The sums of the numbers at the ends of the line segments AB, BC, CD, DE and EA form an arithmetic sequence although not necessarily in that order What is the middle term of the arithmetic sequence?


To solve this problem, we need to understand that the sum of the numbers at the ends of the line segments \( AB, BC, CD, DE, \) and \( EA \) forms an arithmetic sequence. Let's break this down:

### Step 1: Define the arithmetic sequence.
An arithmetic sequence has the property that the difference between consecutive terms is constant. Let the five sums be \( S_1, S_2, S_3, S_4, \) and \( S_5 \). Since these sums form an arithmetic sequence, we can express them as:

\[
S_1, \quad S_1 + d, \quad S_1 + 2d, \quad S_1 + 3d, \quad S_1 + 4d
\]

Where \( d \) is the common difference.

### Step 2: Represent the sums.
Given the numbers 3, 5, 6, 7, and 9, the sums at the ends of the line segments are:

\[
S_1 = A + B, \quad S_2 = B + C, \quad S_3 = C + D, \quad S_4 = D + E, \quad S_5 = E + A
\]

### Step 3: Determine the total sum.
The total sum of these numbers is:

\[
3 + 5 + 6 + 7 + 9 = 30
\]

The sum of all the line segment sums will be double this, since each number is used in exactly two sums:

\[
S_1 + S_2 + S_3 + S_4 + S_5 = 2 \times 30 = 60
\]

### Step 4: Distribute the sums.
Since the sums form an arithmetic sequence:

\[
S_1 + (S_1 + d) + (S_1 + 2d) + (S_1 + 3d) + (S_1 + 4d) = 5S_1 + 10d = 60
\]

### Step 5: Solve for \( S_1 \) and \( d \).
\[
5S_1 + 10d = 60
\]
\[
S_1 + 2d = 12
\]

\( S_1 \) and \( d \) are integers. Now, let's consider possible sums and find the middle term \( S_1 + 2d \) that matches the given numbers.

We have five sums that could be formed by pairs of numbers: 8, 11, 12, 13, 14, and 16.

Possible valid sums for the sequence are \( S_1 = 8, 10, \text{or } 12 \) and we need to find a common difference \( d \) that gives three valid sums corresponding to these sums.

Let’s test \( S_1 = 8 \):

\[
S_1 = 8, \quad S_2 = 10, \quad S_3 = 12, \quad S_4 = 14, \quad S_5 = 16
\]

All of these sums are possible using the numbers 3, 5, 6, 7, and 9.

### Conclusion:
The arithmetic sequence here is \( 8, 10, 12, 14, 16 \), and thus the middle term of this sequence is \( \boxed{12} \).

Would you like to see more details or have any questions about this solution?

Here are five related questions for further exploration:
1. What is the common difference in the arithmetic sequence formed by the sums?
2. How can you arrange the numbers 3, 5, 6, 7, and 9 to get this arithmetic sequence?
3. What if the numbers were 4, 5, 6, 8, and 9? Would the middle term change?
4. How many different arithmetic sequences can you form with these numbers?
5. How does the problem change if the sums are required to be in a geometric sequence instead?

**Tip:** When working with arithmetic sequences, always check the total sum and distribution to ensure all terms fit within the sequence criteria.

In the five-sided star shown, the letters A, B, C, D and E are replaced by the numbers 3,5,6,7, and 9, although not necessarily in this order. The sums of the numbers at the ends of the line segments AB, BC, CD, DE and EA form an arithmetic sequence although not necessarily in that order. What is the middle term of the arithmetic sequence?

Explore how to find the middle term in an arithmetic sequence formed by the sums of numbers placed on the vertices of a five-sided star. This problem involves key mathematical concepts such as arithmetic sequences, algebra, and the sum of pairs. Ideal for students in Grades 8-10.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation