7, 65, 215, 513, 999, ….

Options

A. 888
B. 1000
C. 1001
D. 1729

Explanation

To find the next number in the series: 7, 65, 215, 513, 999, we can analyze the pattern step by step.

### Step 1: Identify the pattern
The given series is:
- 7
- 65
- 215
- 513
- 999

Let's observe the differences between consecutive terms:

1. 65 - 7 = 58
2. 215 - 65 = 150
3. 513 - 215 = 298
4. 999 - 513 = 486

So, the differences are:
- 58, 150, 298, 486

### Step 2: Find the differences of the differences
Next, we can find the differences of these differences:

1. 150 - 58 = 92
2. 298 - 150 = 148
3. 486 - 298 = 188

So, the second differences are:
- 92, 148, 188

### Step 3: Find the differences of the second differences
Now, let's find the differences of the second differences:

1. 148 - 92 = 56
2. 188 - 148 = 40

So, the third differences are:
- 56, 40

### Step 4: Find the differences of the third differences
Now, let's find the differences of the third differences:

1. 40 - 56 = -16

Since we have a constant difference in the third differences, we can assume that the next difference will also decrease by 16.

### Step 5: Calculate the next second difference
If we subtract 16 from the last third difference (40), we get:
40 - 16 = 24

### Step 6: Calculate the next first difference
Now, we add this new second difference (24) to the last second difference (188):
188 + 24 = 212

### Step 7: Calculate the next difference
Now, we add this new first difference (212) to the last first difference (486):
486 + 212 = 698

### Step 8: Calculate the next term in the series
Finally, we add this new difference (698) to the last term in the series (999):
999 + 698 = 1697

Thus, the next number in the series is **1697**.

### Summary of Steps
1. Identify the differences between terms.
2. Find the differences of those differences.
3. Continue finding differences until a pattern emerges.
4. Use the pattern to calculate the next term.