`{:("5","12","30","70","154",),("2","(A)","(B)","(C)","(D)","(E)"):}`
Out of the following, which number will replace (D)?

Options

A. 230
B. 84
C. 46
D. 106

Explanation

To solve the problem, we need to identify the pattern in the first series of numbers (5, 12, 30, 70, 154) and then apply that pattern to the second series (2, (A), (B), (C), (D), (E)) to find the value of (D).

### Step-by-Step Solution:

1. **Identify the Pattern in the First Series:**
- The first series is: 5, 12, 30, 70, 154.
- To find the relationship between the numbers, let's look at how we can go from one number to the next.

2. **Calculate the Differences:**
- From 5 to 12:
- Multiply 5 by 2 = 10, then add 2 → 10 + 2 = 12.
- From 12 to 30:
- Multiply 12 by 2 = 24, then add 6 → 24 + 6 = 30.
- From 30 to 70:
- Multiply 30 by 2 = 60, then add 10 → 60 + 10 = 70.
- From 70 to 154:
- Multiply 70 by 2 = 140, then add 14 → 140 + 14 = 154.

3. **Identify the Adding Pattern:**
- The numbers added are: 2, 6, 10, 14.
- The difference between these additions is consistently 4:
- 6 - 2 = 4
- 10 - 6 = 4
- 14 - 10 = 4

4. **Establish the Pattern:**
- The pattern is to multiply by 2 and then add an increasing sequence of numbers (2, 6, 10, 14) which increases by 4 each time.

5. **Apply the Pattern to the Second Series:**
- The second series starts with 2.
- We need to find (A), (B), (C), (D), and (E).

6. **Calculate (A):**
- Start with 2:
- Multiply 2 by 2 = 4, then add 2 → 4 + 2 = 6.
- So, (A) = 6.

7. **Calculate (B):**
- Start with 6:
- Multiply 6 by 2 = 12, then add 6 → 12 + 6 = 18.
- So, (B) = 18.

8. **Calculate (C):**
- Start with 18:
- Multiply 18 by 2 = 36, then add 10 → 36 + 10 = 46.
- So, (C) = 46.

9. **Calculate (D):**
- Start with 46:
- Multiply 46 by 2 = 92, then add 14 → 92 + 14 = 106.
- So, (D) = 106.

10. **Final Answer:**
- The value that will replace (D) is **106**.