Let's denote the eight numbers as $a_1, a_2, a_3, \ldots, a_8$. The sequence is defined such that each new number is the product of the two previous numbers. The last three numbers provided are:

• $a_6 = 16$
• $a_7 = 64$
• $a_8 = 1024$

We need to find the first number $a_1$.

Given the sequence pattern:

1. $a_8 = a_7 \times a_6$
2. $a_7 = a_6 \times a_5$
3. $a_6 = a_5 \times a_4$

Starting from the given last three numbers:

1. $a_8 = 1024 = a_7 \times a_6 = 64 \times 16$
2. $a_7 = 64 = a_6 \times a_5$

From $a_7 = 64$ and $a_6 = 16$, we find $a_5$:

$a_5 = \frac{a_7}{a_6} = \frac{64}{16} = 4$

Next, using $a_6 = 16$ and $a_5 = 4$, we find $a_4$:

$a_4 = \frac{a_6}{a_5} = \frac{16}{4} = 4$

Continuing this process:

• $a_5 = 4$
• $a_4 = 4$
• $a_3 = \frac{a_5}{a_4} = \frac{4}{4} = 1$
• $a_2 = \frac{a_4}{a_3} = \frac{4}{1} = 4$
• $a_1 = \frac{a_3}{a_2} = \frac{1}{4} = 0.25$

Thus, the first number $a_1$ is $0.25$.

Would you like more details or have any questions?

Here are some questions you might be interested in:

1. How do we verify the calculations for each step?
2. Can you explain why each term is the product of the two previous terms?
3. What would happen if one of the starting numbers were zero?
4. How would the sequence change if we altered the formula?
5. Can you explain the significance of each number in the sequence?

Tip: Always verify your steps when solving problems to ensure accuracy.