Calculate the number of vans needed by dividing the number of people that need transported by the capacity of one van:

vans = \( \frac{82}{7} \) = 11\(\frac{5}{7}\)

So, it will take 11 full vans and one partially full van to transport the entire team making a total of 12 vans.

The equation for this sequence is:

a_n = a_n-1 + 4(n - 1)

where n is the term's order in the sequence, a_n is the value of the term, and a_n-1 is the value of the term before a_n. This makes the next number:

a₆ = a₅ + 4(6 - 1)
a₆ = 41 + 4(5)
a₆ = 61

A rational number (or fraction) is represented as a ratio between two integers, a and b, and has the form \({a \over b}\) where a is the numerator and b is the denominator. An improper fraction (\({5 \over 3} \)) has a numerator with a greater absolute value than the denominator and can be converted into a mixed number (\(1 {2 \over 3} \)) which has a whole number part and a fractional part.

The least common multiple (LCM) is the smallest positive integer that is a multiple of two or more integers.

To subtract these fractions, first find the lowest common multiple of their denominators. The first few multiples of 6 are [6, 12, 18, 24, 30, 36, 42, 48, 54, 60] and the first few multiples of 10 are [10, 20, 30, 40, 50, 60, 70, 80, 90]. The first few multiples they share are [30, 60, 90] making 30 the smallest multiple 6 and 10 share.

Next, convert the fractions so each denominator equals the lowest common multiple:

\( \frac{8 x 5}{6 x 5} \) - \( \frac{5 x 3}{10 x 3} \)

\( \frac{40}{30} \) - \( \frac{15}{30} \)

Now, because the fractions share a common denominator, you can subtract them:

\( \frac{40 - 15}{30} \) = \( \frac{25}{30} \) = \(\frac{5}{6}\)