Step 1: Understanding the Question:
The topic for this problem is Number Series, which is a sub-section of Logical Reasoning and Quantitative Aptitude. A number series is a sequence of numbers following a specific mathematical pattern. To find the missing term, we must analyze the relationship between consecutive numbers (3, 9, 27, 81) to determine the rule that governs the growth of the sequence.
Step 2 : Key Formulas and approach:
1. Geometric Progression (GP) Rule: Each term is obtained by multiplying the previous term by a constant value called the common ratio ($r$).
2. Power Rule: Sometimes series represent sequential powers of a specific base number ($n^1, n^2, n^3 \dots$).
The approach involves checking for an addition pattern first, and if that fails, checking for a multiplication or exponential pattern.
Step 3 : Detailed Explanation:
Let's look at the gap between the terms. $9 - 3 = 6$, $27 - 9 = 18$, $81 - 27 = 54$. Since the difference is increasing rapidly, it is likely not an addition-based series.
Now, let's look for a multiplicative relationship. To get from 3 to 9, we multiply by 3 ($3 \times 3 = 9$).
To get from 9 to 27, we check $9 \times 3$, which indeed equals 27.
To get from 27 to 81, we check $27 \times 3$, which equals 81. This confirms that the common ratio ($r$) is 3.
This sequence also represents the powers of 3: $3^1 = 3$, $3^2 = 9$, $3^3 = 27$, and $3^4 = 81$.
To find the next term, we simply follow the established rule. We must calculate $81 \times 3$ or find the value of $3^5$.
Calculating the product: $80 \times 3 = 240$ and $1 \times 3 = 3$. Adding them together, $240 + 3 = 243$.
This fits perfectly with the exponential progression. The next number in the sequence after 243 would be $243 \times 3 = 729$, but the question only asks for the immediate next term.
Step 4 : Final Answer:
The missing term in the geometric series is 243, which corresponds to option (C).