In each of the following questions a number series is given followed by (a) (b) (c) (d) and (e). You have to complete the series starting with given number, following the sequence of original series and answer the questions that follows the series
3 13 37 87 191 401 1 (a) (b) (c) (d) (e) What will come in place of (d)?

Options

A. 169
B. 161
C. 171
D. 159

Answer

To solve the given number series and find the value that will come in place of (d), we will analyze the series step by step.

### Given Series:
3, 13, 37, 87, 191, 401, 1, (a), (b), (c), (d), (e)

### Step 1: Identify the Pattern
Let's look at the differences between consecutive terms in the series:

- From 3 to 13:
\( 13 - 3 = 10 \)

- From 13 to 37:
\( 37 - 13 = 24 \)

- From 37 to 87:
\( 87 - 37 = 50 \)

- From 87 to 191:
\( 191 - 87 = 104 \)

- From 191 to 401:
\( 401 - 191 = 210 \)

### Step 2: Analyze the Differences
Now, let's analyze the differences we calculated:

- Differences: 10, 24, 50, 104, 210

Next, we will find the differences of these differences:

- From 10 to 24:
\( 24 - 10 = 14 \)

- From 24 to 50:
\( 50 - 24 = 26 \)

- From 50 to 104:
\( 104 - 50 = 54 \)

- From 104 to 210:
\( 210 - 104 = 106 \)

### Step 3: Analyze the Second Differences
Now, let's analyze the second differences:

- Second Differences: 14, 26, 54, 106

Now, we will find the differences of these second differences:

- From 14 to 26:
\( 26 - 14 = 12 \)

- From 26 to 54:
\( 54 - 26 = 28 \)

- From 54 to 106:
\( 106 - 54 = 52 \)

### Step 4: Identify the Pattern in Differences
The differences of the second differences are:

- Differences of Second Differences: 12, 28, 52

This shows a pattern of increasing differences.

### Step 5: Predict the Next Terms
Continuing the pattern, we can predict the next difference:

- The next difference of the second differences can be calculated by observing the pattern. The increments seem to be increasing by 16 each time (12 + 16 = 28, 28 + 24 = 52).

Thus, the next difference could be \( 52 + 16 = 68 \).

Now, we can add this to the last second difference:

- Next second difference: \( 106 + 68 = 174 \)

Now, add this to the last first difference:

- Next first difference: \( 210 + 174 = 384 \)

### Step 6: Calculate the Next Term
Now, we can find the next term in the series:

- Next term after 401:
\( 401 + 384 = 785 \)

### Step 7: Continue the Series
Now, we can repeat this process to find the next terms (a), (b), (c), (d), and (e):

1. From 785, calculate the next first difference:
- Next first difference: \( 384 + 174 = 558 \)
- Next term: \( 785 + 558 = 1343 \)

2. From 1343, calculate the next first difference:
- Next first difference: \( 558 + 174 = 732 \)
- Next term: \( 1343 + 732 = 2075 \)

3. From 2075, calculate the next first difference:
- Next first difference: \( 732 + 174 = 906 \)
- Next term: \( 2075 + 906 = 2981 \)

4. From 2981, calculate the next first difference:
- Next first difference: \( 906 + 174 = 1080 \)
- Next term: \( 2981 + 1080 = 4061 \)

### Conclusion
The series continues as follows:
- 3, 13, 37, 87, 191, 401, 785, (a = 1343), (b = 2075), (c = 2981), (d = 4061), (e = ...)

Thus, the value that will come in place of (d) is **4061**.