1 5 13 29 ? 125 253

Options

A. 83
B. 69
C. 61
D. 65

Explanation

To solve the series: 1, 5, 13, 29, ?, 125, 253, we will analyze the differences between consecutive terms to identify a pattern.

### Step 1: Identify the differences between consecutive terms
- The difference between the first term (1) and the second term (5) is:
\[ 5 - 1 = 4 \]
- The difference between the second term (5) and the third term (13) is:
\[ 13 - 5 = 8 \]
- The difference between the third term (13) and the fourth term (29) is:
\[ 29 - 13 = 16 \]

### Step 2: Observe the pattern in the differences
Now we have the differences:
- 4, 8, 16

Notice that these differences are powers of 2:
- \(2^2 = 4\)
- \(2^3 = 8\)
- \(2^4 = 16\)

### Step 3: Predict the next difference
Following the pattern, the next difference should be:
- \(2^5 = 32\)

### Step 4: Calculate the missing term
Now we add this difference to the last known term (29):
\[ 29 + 32 = 61 \]

### Step 5: Verify the next terms in the series
Now let's check the next terms in the series:
- The difference between 61 and 125:
\[ 125 - 61 = 64 \quad (which is \, 2^6) \]
- The difference between 125 and 253:
\[ 253 - 125 = 128 \quad (which is \, 2^7) \]

### Conclusion
The missing term in the series is:
\[ \boxed{61} \]