A ratio of 4:1 means that there are 4 home fans for every one visiting fan. So, of every 5 fans, 4 are home fans and \( \frac{4}{5} \) of every fan in the stadium is a home fan:

38,000 fans x \( \frac{4}{5} \) = \( \frac{152000}{5} \) = 30,400 fans.

To add these fractions, first find the lowest common multiple of their denominators. The first few multiples of 2 are [2, 4, 6, 8, 10, 12, 14, 16, 18, 20] and the first few multiples of 10 are [10, 20, 30, 40, 50, 60, 70, 80, 90]. The first few multiples they share are [10, 20, 30, 40, 50] making 10 the smallest multiple 2 and 10 share.

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

\( \frac{7 x 5}{2 x 5} \) + \( \frac{4 x 1}{10 x 1} \)

\( \frac{35}{10} \) + \( \frac{4}{10} \)

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

\( \frac{35 + 4}{10} \) = \( \frac{39}{10} \) = 3\(\frac{9}{10}\)

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

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

4 + (5 + 3) ÷ 4 x 4 - 4²
P: 4 + (8) ÷ 4 x 4 - 4²
E: 4 + 8 ÷ 4 x 4 - 16
MD: 4 + \( \frac{8}{4} \) x 4 - 16
MD: 4 + \( \frac{32}{4} \) - 16
AS: \( \frac{16}{4} \) + \( \frac{32}{4} \) - 16
AS: \( \frac{48}{4} \) - 16
AS: \( \frac{48 - 64}{4} \)
\( \frac{-16}{4} \)
-4

The equation for this sequence is:

a_n = a_n-1 + 2

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 + 2
a₆ = 9 + 2
a₆ = 11