What should come in place of the question mark (?) in the following number series?
`7" "12" "32" "105" "?`

Options

A. A)428
B. B)241
C. C)218
D. D)416

Answer

To find the missing term in the series `7, 12, 32, 105, ?`, we will analyze the pattern in the series step by step.

### Step 1: Identify the pattern
We will look at how each term relates to the previous term.

1. **From 7 to 12**:
- We can see that \( 7 + 5 = 12 \).

2. **From 12 to 32**:
- Here, \( 12 + (4 \times 5) = 12 + 20 = 32 \).

3. **From 32 to 105**:
- For this step, \( 32 + (3 \times 24) = 32 + 72 = 105 \).

### Step 2: Establish the pattern
Looking closely, we can see that:
- The first term (7) is added to 5 (which is \( 5 \times 1 \)).
- The second term (12) is added to \( 4 \times 5 \) to get the third term (32).
- The third term (32) is added to \( 3 \times 24 \) to get the fourth term (105).

### Step 3: Determine the next term
Following the established pattern, we can see that:
- The addition values are decreasing by 1 each time (5, 4, 3).
- The multiplier for the last term (105) was 3, so for the next term, we will multiply by 4.

### Step 4: Calculate the missing term
To find the fifth term:
- We will add \( 2 \times 105 \) to 105:
\[ 105 + (2 \times 105) = 105 + 210 = 315 \]

### Step 5: Finalize the answer
Now, we will check the calculations:
- 7 to 12: \( 7 + 5 = 12 \)
- 12 to 32: \( 12 + (4 \times 5) = 32 \)
- 32 to 105: \( 32 + (3 \times 24) = 105 \)
- 105 to ?: \( 105 + (2 \times 105) = 315 \)

Thus, the missing term in the series is **315**.