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.