find the sum of all numbers multiple of 6 between 10 and 90

To find the sum of all numbers that are multiples of 6 between 10 and 90, follow these steps:

1. **Identify the multiples of 6 within the range**:
   - The smallest multiple of 6 greater than 10 is 12.
   - The largest multiple of 6 less than or equal to 90 is 90.

   So, the multiples of 6 in this range are:  
   \( 12, 18, 24, 30, 36, 42, 48, 54, 60, 66, 72, 78, 84, 90 \).

2. **Calculate the sum of these multiples**:
   The sum of an arithmetic series can be calculated using the formula:
   \[
   S_n = \frac{n}{2} \times (a + l)
   \]
   Where:
   - \( S_n \) is the sum of the series,
   - \( n \) is the number of terms,
   - \( a \) is the first term,
   - \( l \) is the last term.

   - \( a = 12 \)
   - \( l = 90 \)
   - To find \( n \) (the number of terms):
     \[
     n = \frac{l - a}{d} + 1 = \frac{90 - 12}{6} + 1 = \frac{78}{6} + 1 = 13 + 1 = 14
     \]

3. **Plug in the values**:
   \[
   S_{14} = \frac{14}{2} \times (12 + 90) = 7 \times 102 = 714
   \]

Thus, the sum of all numbers that are multiples of 6 between 10 and 90 is **714**.

Would you like further details or have any questions?

Here are five related questions:
1. What is the sum of all multiples of 5 between 10 and 100?
2. How many multiples of 6 are there between 10 and 90?
3. What is the average of all multiples of 6 between 10 and 90?
4. What is the sum of the squares of all multiples of 6 between 10 and 90?
5. What is the sum of all numbers multiple of 7 between 20 and 70?

**Tip:** When working with arithmetic sequences, knowing how to quickly identify the first and last terms can make calculations much simpler.

Learn how to find the sum of all numbers that are multiples of 6 between 10 and 90 using arithmetic sequences and the sum formula. This problem covers key mathematical concepts like arithmetic sequences and series summation. Suitable for students in Grades 7-9.

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