Understanding the Number Series

The question presents a sequence of numbers: 729, 512, 343, ?, 125. We need to find the missing number that fits the pattern.

Identifying the Pattern: Perfect Cubes

Let's examine each number in the series to see if there's a recognizable pattern:

• The first number is 729. This can be expressed as the cube of 9: $9^3 = 9 \times 9 \times 9 = 729$.
• The second number is 512. This is the cube of 8: $8^3 = 8 \times 8 \times 8 = 512$.
• The third number is 343. This is the cube of 7: $7^3 = 7 \times 7 \times 7 = 343$.
• The fifth number is 125. This is the cube of 5: $5^3 = 5 \times 5 \times 5 = 125$.

Observing these values, we can see that the series consists of consecutive integers cubed, starting from 9 and decreasing:

$9^3$, $8^3$, $7^3$, ?, $5^3$

Calculating the Missing Number

Following the established pattern of descending integer cubes, the missing number should be the cube of 6.

Calculation:

$6^3 = 6 \times 6 \times 6$

$6^3 = 36 \times 6$

$6^3 = 216$

So, the missing number in the series is 216.

Confirming the Series

The complete series, including the missing number, is:

$9^3, 8^3, 7^3, 6^3, 5^3$

Which translates to:

729, 512, 343, 216, 125

The number 216 fits perfectly into the sequence.

Selecting the Correct Option

Comparing our calculated missing number (216) with the given options:

• 1. 16
• 2. 64
• 3. 576
• 4. 216

The number 216 matches option 4.