The equation for this sequence is:

a_n = a_n-1 + 9

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 + 9
a₆ = 37 + 9
a₆ = 46

43% of the water consumption is industrial use and 20% is personal use so (43% - 20%) = 23% more water is used for industrial purposes. 10,000 gallons are consumed daily so industry consumes \( \frac{23}{100} \) x 10,000 gallons = 2,300 gallons.

Use PEMDAS (Parentheses, Exponents, Multipy/Divide, Add/Subtract):

4 + (3 + 3) ÷ 4 x 2 - 2²
P: 4 + (6) ÷ 4 x 2 - 2²
E: 4 + 6 ÷ 4 x 2 - 4
MD: 4 + \( \frac{6}{4} \) x 2 - 4
MD: 4 + \( \frac{12}{4} \) - 4
AS: \( \frac{16}{4} \) + \( \frac{12}{4} \) - 4
AS: \( \frac{28}{4} \) - 4
AS: \( \frac{28 - 16}{4} \)
\( \frac{12}{4} \)
3

A ratio of 2:1 means that there are 2 home fans for every one visiting fan. So, of every 3 fans, 2 are home fans and \( \frac{2}{3} \) of every fan in the stadium is a home fan:

43,000 fans x \( \frac{2}{3} \) = \( \frac{86000}{3} \) = 28,667 fans.

To multiply terms with radicals, multiply the coefficients and radicands separately:

2\( \sqrt{8} \) x 4\( \sqrt{5} \)
(2 x 4)\( \sqrt{8 \times 5} \)
8\( \sqrt{40} \)

Now we need to simplify the radical:

8\( \sqrt{40} \)
8\( \sqrt{10 \times 4} \)
8\( \sqrt{10 \times 2^2} \)
(8)(2)\( \sqrt{10} \)
16\( \sqrt{10} \)