To find the sum of all positive multiples of 7 that are less than 90, we first identify those multiples.

The positive multiples of 7 that are less than 90 are: $7, 14, 21, 28, 35, 42, 49, 56, 63, 70, 77, 84$

This sequence forms an arithmetic series where:

• The first term $a_1 = 7$
• The common difference $d = 7$
• The last term $a_n = 84$

The number of terms $n$ in this arithmetic sequence can be calculated using the formula for the $n$-th term of an arithmetic sequence: $a_n = a_1 + (n-1) \times d$ Plugging in the values: $84 = 7 + (n-1) \times 7$ $84 - 7 = (n-1) \times 7$ $77 = (n-1) \times 7$ $n-1 = 11$ $n = 12$ So, there are 12 terms.

The sum $S_n$ of an arithmetic series can be calculated using the formula: $S_n = \frac{n}{2} \times (a_1 + a_n)$ Substituting the known values: $S_{12} = \frac{12}{2} \times (7 + 84)$ $S_{12} = 6 \times 91$ $S_{12} = 546$

So, the sum of all the positive multiples of 7 that are less than 90 is $\boxed{546}$.

Would you like to go over any details, or do you have any questions?

Here are five related questions:

1. How would you find the sum of multiples of 7 that are less than 150?
2. Can you find the sum of all multiples of 5 that are less than 100?
3. How would you modify the method if asked for multiples of a different number?
4. What would be the sum of all positive multiples of 8 less than 80?
5. Can you determine the average of the multiples of 7 less than 90?

Tip: When working with arithmetic sequences, always double-check the number of terms and common difference to ensure accuracy.