Number Series Pattern Recognition

The question asks us to identify the missing number in the given sequence: $2, \ 4, \ 8, \ 16, \ ?, \ 64$. To solve this, we need to find the underlying pattern connecting the numbers in the series.

Analyzing the Series Pattern

Let's examine the relationship between consecutive terms:

• The first term is 2.
• The second term is 4. The relationship between the first and second term is $2 \times 2 = 4$.
• The third term is 8. The relationship between the second and third term is $4 \times 2 = 8$.
• The fourth term is 16. The relationship between the third and fourth term is $8 \times 2 = 16$.

The pattern observed is that each term is obtained by multiplying the previous term by 2. This indicates a geometric progression with a common ratio of 2.

Calculating the Missing Number

Following this pattern, the missing number (the fifth term) should be the fourth term multiplied by 2:

Missing Number = $16 \times 2$

Missing Number = $32$

Verifying the Pattern

To ensure the pattern holds true for the entire series, let's calculate the next term using the identified missing number:

The term after the missing number is 64. If the missing number is 32, then:

$32 \times 2 = 64$

This matches the last term in the series, confirming that the pattern is consistent.

Conclusion

The missing number in the series $2, \ 4, \ 8, \ 16, \ ?, \ 64$ is 32. This corresponds to option 3.