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

To multiply fractions, multiply the numerators together and then multiply the denominators together:

\( \frac{2}{7} \) x \( \frac{2}{5} \) = \( \frac{2 x 2}{7 x 5} \) = \( \frac{4}{35} \) = \(\frac{4}{35}\)

30% of the water consumption is industrial use and 18% is personal use so (30% - 18%) = 12% more water is used for industrial purposes. 5,000 gallons are consumed daily so industry consumes \( \frac{12}{100} \) x 5,000 gallons = 600 gallons.

A factorial has the form n! and is the product of the integer (n) and all the positive integers below it. For example, 5! = 5 x 4 x 3 x 2 x 1 = 120.

To add these fractions, first find the lowest common multiple of their denominators. The first few multiples of 8 are [8, 16, 24, 32, 40, 48, 56, 64, 72, 80] and the first few multiples of 12 are [12, 24, 36, 48, 60, 72, 84, 96]. The first few multiples they share are [24, 48, 72, 96] making 24 the smallest multiple 8 and 12 share.

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

\( \frac{4 x 3}{8 x 3} \) + \( \frac{2 x 2}{12 x 2} \)

\( \frac{12}{24} \) + \( \frac{4}{24} \)

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

\( \frac{12 + 4}{24} \) = \( \frac{16}{24} \) = \(\frac{2}{3}\)