16 , 33,65 ,131,261 (…)

Options

A. 523
B. 521
C. 613
D. 721

Answer

To find the next term in the series 16, 33, 65, 131, 261, we will analyze the pattern of the series step by step.

### Step 1: Identify the pattern
Let's look at the differences between consecutive terms:

- From 16 to 33:
\( 33 - 16 = 17 \)

- From 33 to 65:
\( 65 - 33 = 32 \)

- From 65 to 131:
\( 131 - 65 = 66 \)

- From 131 to 261:
\( 261 - 131 = 130 \)

So, the differences are:
17, 32, 66, 130.

### Step 2: Analyze the differences
Now, let's look at the differences of these differences:

- From 17 to 32:
\( 32 - 17 = 15 \)

- From 32 to 66:
\( 66 - 32 = 34 \)

- From 66 to 130:
\( 130 - 66 = 64 \)

So, the second differences are:
15, 34, 64.

### Step 3: Analyze the second differences
Now, let's look at the differences of these second differences:

- From 15 to 34:
\( 34 - 15 = 19 \)

- From 34 to 64:
\( 64 - 34 = 30 \)

So, the third differences are:
19, 30.

### Step 4: Predict the next difference
Now, let's analyze the differences of these third differences:

- From 19 to 30:
\( 30 - 19 = 11 \)

This suggests that the next difference might increase by 11 again, giving us:
\( 30 + 11 = 41 \).

### Step 5: Calculate the next second difference
Adding this to the last second difference:
\( 64 + 41 = 105 \).

### Step 6: Calculate the next first difference
Adding this to the last first difference:
\( 130 + 105 = 235 \).

### Step 7: Calculate the next term in the series
Now, we add this to the last term in the original series:
\( 261 + 235 = 496 \).

Thus, the next term in the series is **496**.

### Summary of the Solution
The next term in the series 16, 33, 65, 131, 261 is **496**.

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