The missing number in the given sequence 343, 1331, ________, 4913 is

Sequence Analysis: Finding the Missing Number

We are asked to find the missing number in the sequence: 343, 1331, ________, 4913.

Let's examine the given numbers to identify a pattern.

Identifying the Pattern

We can observe that the numbers in the sequence might be related to powers or specific mathematical operations.

- The first number is 343. We can recognize this as $7\times 7\times 7$, or $7^3$.
- The second number is 1331. This can be calculated as $11\times 11\times 11$, or $11^3$.
- The fourth number is 4913. Let's check if this is a cube of a prime number. Testing $17^3$, we get $17\times 17\times 17=289\times 17=4913$. So, the fourth number is $17^3$.

The sequence of bases is 7, 11, ?, 17.

Determining the Missing Base

Let's look at the sequence of bases: 7, 11, ?, 17. These numbers appear to be prime numbers.

The sequence of prime numbers starts as: 2, 3, 5, 7, 11, 13, 17, 19, ...

Comparing our bases (7, 11, ?, 17) with the prime number sequence, we see that 7 and 11 are consecutive primes. The next prime number after 11 is 13, and the prime number after 13 is 17. This fits perfectly with the given sequence.

Therefore, the missing base is 13.

Calculating the Missing Number

Since the pattern involves cubes of prime numbers, the missing number is $13^3$.

Calculation:

- $13^3=13\times 13\times 13$
- $13\times 13=169$
- $169\times 13=(100+60+9)\times 13$
- $=100\times 13+60\times 13+9\times 13$
- $=1300+780+117$
- $=2080+117$
- $=2197$

So, the missing number in the sequence is 2197.

Verifying the Sequence

The complete sequence based on the identified pattern is:

- $7^3=343$
- $11^3=1331$
- $13^3=2197$
- $17^3=4913$

The sequence is formed by cubing consecutive prime numbers starting from 7 (7, 11, 13, 17).