A number series is given with one term missing. Choose the correct alternative from the given options that will complete the series. 7, 49, 343, ?

Introduction / Context:
This question involves a geometric style series where each term is obtained by multiplying the previous term by the same integer. In this example, the base number is seven, so the series progresses through powers of seven. Identifying such power patterns is a useful skill in many competitive examinations.

Given Data / Assumptions:
- Series: 7, 49, 343, ? - One term is missing in the fourth position. - Terms appear to grow rapidly, suggesting multiplication by a fixed number.

Concept / Approach:
When each term is obtained by multiplying the previous term by a constant factor, the series is geometric. Here, the factor appears to be 7. Recognising that 7, 49, and 343 are 7^1, 7^2, and 7^3 helps to project the next term as 7^4. This power based understanding gives a fast and reliable method for finding the missing value.

Step-by-Step Solution:
Step 1: Express each term as a power of seven: 7 = 7^1, 49 = 7^2, 343 = 7^3. Step 2: Observe that exponents increase by one at each step: 1, 2, 3. Step 3: The next exponent should be 4, so the next term is 7^4. Step 4: Compute 7^4 as 7 * 7 * 7 * 7 = 49 * 49 = 2401. Step 5: Therefore, the missing term equals 2401.

Verification / Alternative check:
Another way is to check the ratio between consecutive terms. 49 / 7 = 7 and 343 / 49 = 7. Thus, the common ratio is 7. Multiplying the third term by this ratio gives 343 * 7 = 2401, which confirms the result and the geometric nature of the series.

Why Other Options Are Wrong:
- Option 3087 is not a power of seven and does not follow the repeated multiplication rule with factor 7. - Option 1029 is 343 * 3, so it uses the wrong multiplier and breaks the geometric pattern. - Option 1091 has no simple relation to the previous term under the constant ratio rule.

Common Pitfalls:
Some learners may attempt to apply addition or subtraction instead of multiplicative reasoning, which will not work for rapidly growing series like this. Others may miscalculate powers of seven, especially 7^4. It helps to remember key powers for small primes, such as 2, 3, 5, and 7, for quick recognition in exams.

Final Answer:
The missing term is 2401, so the correct option is 2401.