The next number in the sequence 2, 8, 18, 32, 50, .... is

Options

A. 68
B. 72
C. 76
D. 80

Answer

To find the next number in the sequence 2, 8, 18, 32, 50, we will analyze the differences between consecutive terms.

**Step 1: Identify the sequence.**
The given sequence is:
2, 8, 18, 32, 50

**Step 2: Calculate the differences between consecutive terms.**
- Difference between 8 and 2:
\( 8 - 2 = 6 \)
- Difference between 18 and 8:
\( 18 - 8 = 10 \)
- Difference between 32 and 18:
\( 32 - 18 = 14 \)
- Difference between 50 and 32:
\( 50 - 32 = 18 \)

So, the differences are:
6, 10, 14, 18

**Step 3: Analyze the differences.**
Now, let's look at the differences we calculated:
- The first difference is 6
- The second difference is 10
- The third difference is 14
- The fourth difference is 18

**Step 4: Identify the pattern in the differences.**
The differences themselves form a sequence:
6, 10, 14, 18

We can see that each difference increases by 4:
- Difference between 10 and 6: \( 10 - 6 = 4 \)
- Difference between 14 and 10: \( 14 - 10 = 4 \)
- Difference between 18 and 14: \( 18 - 14 = 4 \)

**Step 5: Predict the next difference.**
Following the pattern, the next difference should be:
\( 18 + 4 = 22 \)

**Step 6: Calculate the next term in the original sequence.**
To find the next term in the original sequence, we add this next difference to the last term (50):
\( 50 + 22 = 72 \)

Thus, the next number in the sequence is **72**.