A factorial is the product of an integer and all the positive integers below it. To solve a fraction featuring factorials, expand the factorials and cancel out like numbers:

\( \frac{4!}{3!} \)
\( \frac{4 \times 3 \times 2 \times 1}{3 \times 2 \times 1} \)
\( \frac{4}{1} \)
4

The equation for this sequence is:

a_n = a_n-1 + 2(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 + 2(6 - 1)
a₆ = 21 + 2(5)
a₆ = 31

To divide fractions, invert the second fraction and then multiply:

\( \frac{4}{7} \) ÷ \( \frac{2}{7} \) = \( \frac{4}{7} \) x \( \frac{7}{2} \)

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

\( \frac{4}{7} \) x \( \frac{7}{2} \) = \( \frac{4 x 7}{7 x 2} \) = \( \frac{28}{14} \) = 2

First, solve for \( \left|b + 1\right| \):

\( \left|b + 1\right| \) + 7 = 2
\( \left|b + 1\right| \) = 2 - 7
\( \left|b + 1\right| \) = -5

The value inside the absolute value brackets can be either positive or negative so (b + 1) must equal - 5 or --5 for \( \left|b + 1\right| \) to equal -5:

So, b = 4 or b = -6.

To add these fractions, first find the lowest common multiple of their denominators. The first few multiples of 4 are [4, 8, 12, 16, 20, 24, 28, 32, 36, 40] and the first few multiples of 8 are [8, 16, 24, 32, 40, 48, 56, 64, 72, 80]. The first few multiples they share are [8, 16, 24, 32, 40] making 8 the smallest multiple 4 and 8 share.

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

\( \frac{8 x 2}{4 x 2} \) + \( \frac{9 x 1}{8 x 1} \)

\( \frac{16}{8} \) + \( \frac{9}{8} \)

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

\( \frac{16 + 9}{8} \) = \( \frac{25}{8} \) = 3\(\frac{1}{8}\)