See the number pattern given below : 1, 8, 27, 64, 125,_____, ______
What will be the next two terms ?

Question

Accepted Answer

The given number pattern is: 1, 8, 27, 64, 125,_____, ______

To determine the next two terms in the sequence, we need to identify the pattern or rule governing the sequence. Let's analyze the given numbers:

1. The first term is 1, which is $1^3$.
2. The second term is 8, which is $2^3$.
3. The third term is 27, which is $3^3$.
4. The fourth term is 64, which is $4^3$.
5. The fifth term is 125, which is $5^3$.

From this analysis, we can see that each term in the sequence is the cube of consecutive natural numbers. Therefore, the pattern is that each term is the cube of its position in the sequence.

Following this pattern:
- The sixth term should be $6^3 = 216$.
- The seventh term should be $7^3 = 343$.

Thus, the next two terms in the sequence are 216 and 343.

Now, let's evaluate why Option 4 is the correct answer and why the other options are incorrect:

1. **Option 1: 256 and 289**
   - 256 is $4^4$ and 289 is $17^2$. These numbers do not follow the pattern of cubes of consecutive natural numbers.

2. **Option 2: 216 and 289**
   - While 216 is correct as $6^3$, 289 is $17^2$, which does not fit the pattern of cubes.

3. **Option 3: 256 and 343**
   - 256 is $4^4$, which does not fit the pattern. Although 343 is correct as $7^3$, the first number in this option is incorrect.

4. **Option 4: 216 and 343**
   - Both numbers fit the pattern perfectly: 216 is $6^3$ and 343 is $7^3$. This option correctly follows the sequence of cubes of consecutive natural numbers.

Therefore, Option 4 is the correct answer as it accurately continues the pattern of the sequence.