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

The first few multiples of 2 are [2, 4, 6, 8, 10, 12, 14, 16, 18, 20] 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 2 and 8 have in common.

The hourly error rate for this machine is the error rate in parts per 100 multiplied by the number of parts produced per hour:

\( \frac{2}{100} \) x 10 = \( \frac{2 \times 10}{100} \) = \( \frac{20}{100} \) = 0.2 errors per hour

So, in an average hour, the machine will produce 10 - 0.2 = 9.8 error free parts.

The machine ran for 24 - 6 = 18 hours yesterday so you would expect that 18 x 9.8 = 176.4 error free parts were produced yesterday.

To add these fractions, first find the lowest common multiple of their denominators. The first few multiples of 5 are [5, 10, 15, 20, 25, 30, 35, 40, 45, 50] and the first few multiples of 7 are [7, 14, 21, 28, 35, 42, 49, 56, 63, 70]. The first few multiples they share are [35, 70] making 35 the smallest multiple 5 and 7 share.

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

\( \frac{4 x 7}{5 x 7} \) + \( \frac{6 x 5}{7 x 5} \)

\( \frac{28}{35} \) + \( \frac{30}{35} \)

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

\( \frac{28 + 30}{35} \) = \( \frac{58}{35} \) = 1\(\frac{23}{35}\)

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:

\( \frac{5!}{4!} \)
\( \frac{5 \times 4 \times 3 \times 2 \times 1}{4 \times 3 \times 2 \times 1} \)
\( \frac{5}{1} \)
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