Question

Find the missing number in the following series: \(4,4,6,12,\ ?,\ 90,315\).

• A. \(24\)
• B. \(26\)
• C. \(30\)
• D. \(32\)

Hint:

In number series, check whether multiplication factors are increasing regularly.

Answer 1

The Correct Option is C

Step 1: Understanding the Question:
The task is to find the mathematical pattern governing the given sequence and calculate the missing term.

Step 2: Key Formula or Approach:
Let us check the ratio between successive terms to see if the series is formed by multiplying by a sequence of increasing numbers.

Step 3: Detailed Explanation:
Let the terms be $T_1, T_2, T_3, T_4, T_5, T_6, T_7$.
Let us find the multipliers between consecutive terms:
• $\frac{T_2}{T_1} = \frac{4}{4} = 1 \implies T_2 = T_1 \times 1$

• $\frac{T_3}{T_2} = \frac{6}{4} = 1.5 \implies T_3 = T_2 \times 1.5$

• $\frac{T_4}{T_3} = \frac{12}{6} = 2 \implies T_4 = T_3 \times 2$
We notice the multiplying factor increases by $0.5$ at each step: $1, 1.5, 2, \dots$
Following this arithmetic progression for the multipliers:
• The next multiplier should be $2 + 0.5 = 2.5$.
Therefore, $T_5$ (the missing term) should be: \[ T_5 = T_4 \times 2.5 = 12 \times 2.5 = 30 \]
• Let us verify this with the subsequent terms: \[ T_6 = T_5 \times 3.0 = 30 \times 3 = 90 \] \[ T_7 = T_6 \times 3.5 = 90 \times 3.5 = 315 \] Both checks are completely consistent with the given series. Thus, the missing number is 30.

Step 4: Final Answer:
The missing number is 30, which corresponds to option (C).