the average of all the numbers between 6 and 50 which are divisible by 5 is 
give a shortcut trick or formula to solve 

To find the average of all numbers between 6 and 50 that are divisible by 5, we can use the following approach:

### Steps to Solve:

1. **Identify the Numbers Divisible by 5**:
   - The numbers between 6 and 50 that are divisible by 5 are: 10, 15, 20, 25, 30, 35, 40, 45, and 50.
   - These numbers form an arithmetic sequence where the first term (\(a_1\)) is 10, and the common difference (\(d\)) is 5.

2. **Count the Numbers**:
   - To count the numbers in the sequence, use the formula for the \(n\)-th term of an arithmetic sequence:
     \[
     a_n = a_1 + (n-1) \cdot d
     \]
     Set \(a_n = 50\) to find \(n\):
     \[
     50 = 10 + (n-1) \cdot 5
     \]
     \[
     40 = (n-1) \cdot 5
     \]
     \[
     n-1 = 8 \quad \Rightarrow \quad n = 9
     \]

3. **Find the Sum of the Numbers**:
   - The sum \(S\) of an arithmetic sequence can be calculated by:
     \[
     S = \frac{n}{2} \times (a_1 + a_n)
     \]
     \[
     S = \frac{9}{2} \times (10 + 50) = \frac{9}{2} \times 60 = 9 \times 30 = 270
     \]

4. **Calculate the Average**:
   - The average \(A\) is the sum of the numbers divided by the count of the numbers:
     \[
     A = \frac{S}{n} = \frac{270}{9} = 30
     \]

### Shortcut Formula:

For any arithmetic sequence of numbers divisible by \(d\) between two numbers \(m\) and \(n\), where \(a_1\) is the first term and \(a_k\) is the last term:

- **Formula**: 
  \[
  A = \frac{a_1 + a_k}{2}
  \]

- **Explanation**: This formula works because the average of an arithmetic sequence is equal to the average of the first and last terms.

### Applying the Shortcut:

In this problem, \(a_1 = 10\) and \(a_k = 50\), so:

\[
A = \frac{10 + 50}{2} = \frac{60}{2} = 30
\]

Thus, the average of all numbers between 6 and 50 divisible by 5 is **30**.

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Do you want any further details or have any questions?

Here are some related questions that might interest you:

1. What is an arithmetic sequence, and how do you identify it?
2. How can you find the sum of an arithmetic sequence using the formula?
3. Can this method be applied to find the average of numbers divisible by another number?
4. How do you determine the number of terms in an arithmetic sequence?
5. What are some real-life applications of arithmetic sequences?
6. How would you find the average of numbers between two given numbers if they are divisible by different values?
7. How can you verify your result using a different method?
8. How does this method differ from finding the average of non-arithmetic sequences?

**Tip**: When dealing with arithmetic sequences, always look for patterns and use formulas to simplify calculations!

Learn a shortcut formula to find the average of all numbers between 6 and 50 that are divisible by 5. Includes step-by-step instructions, formulas for arithmetic sequences and averages, and practical applications. Suitable for Grades 7-9.

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