The equation for this sequence is:

a_n = a_n-1 + 3(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 + 3(6 - 1)
a₆ = 31 + 3(5)
a₆ = 46

The absolute value is the positive magnitude of a particular number or variable and is indicated by two vertical lines: \(\left|-5\right| = 5\). In the case of a variable absolute value (\(\left|a\right| = 5\)) the value of a can be either positive or negative (a = -5 or a = 5).

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

vans = \( \frac{84}{11} \) = 7\(\frac{7}{11}\)

So, it will take 7 full vans and one partially full van to transport the entire team making a total of 8 vans.

There are 3 2-passenger taxis available so that's 3 x 2 = 6 total seats. There are 11 people needing transportation leaving 11 - 6 = 5 who will have to find other transportation.

If a single cook can bake 2 large cakes per hour and the kitchen is available for 4 hours, a single cook can bake 2 x 4 = 8 large cakes during that time. 37 large cakes are needed for the party so \( \frac{37}{8} \) = 4\(\frac{5}{8}\) cooks are needed to bake the required number of large cakes.

If a single cook can bake 19 small cakes per hour and the kitchen is available for 4 hours, a single cook can bake 19 x 4 = 76 small cakes during that time. 360 small cakes are needed for the party so \( \frac{360}{76} \) = 4\(\frac{14}{19}\) cooks are needed to bake the required number of small cakes.

Because you can't employ a fractional cook, round the number of cooks needed for each type of cake up to the next whole number resulting in 5 + 5 = 10 cooks.