Solving the Number Series Puzzle: 4, 27, 16, 125, ?, 343

This problem requires us to identify a pattern in the given number series and find the missing term represented by the question mark (?). The series is: 4, 27, 16, 125, ?, 343.

Analyzing the Number Series Pattern

Let's examine the numbers in the series. We have:

• Position 1: 4
• Position 2: 27
• Position 3: 16
• Position 4: 125
• Position 5: ?
• Position 6: 343

Observing the numbers, we can see they might relate to powers (squares and cubes). Let's try splitting the series into two separate sequences based on their position (odd or even):

Odd Position Sequence

The numbers at odd positions (1st, 3rd, 5th) are: 4, 16, ?

• The 1st term is 4, which can be written as $2^2$.
• The 3rd term is 16, which can be written as $4^2$.

The bases of these squares are 2 and 4. It appears the bases are increasing by 2 for each subsequent odd position (2, 4, ...). Following this pattern, the base for the 5th term (the next odd position) should be 6. The exponent remains 2.

Therefore, the 5th term should be $6^2$.

Even Position Sequence

The numbers at even positions (2nd, 4th, 6th) are: 27, 125, 343

• The 2nd term is 27, which can be written as $3^3$.
• The 4th term is 125, which can be written as $5^3$.
• The 6th term is 343, which can be written as $7^3$.

The bases of these cubes are 3, 5, and 7. These are consecutive odd numbers. The exponent remains 3.

Calculating the Missing Term

Based on the identified patterns:

• The odd positions follow the pattern $n^2$, where n increases as 2, 4, 6,...
• The even positions follow the pattern $m^3$, where m increases as 3, 5, 7,...

The missing term is at the 5th position, which is an odd position.

Using the pattern for odd positions, the base is 6 and the exponent is 2.

So, the missing number is $6^2 = 36$.

The complete series, following this dual pattern, is: $2^2$, $3^3$, $4^2$, $5^3$, $6^2$, $7^3$, which corresponds to 4, 27, 16, 125, 36, 343.

Thus, the number that replaces the question mark (?) is 36.