The next term of the sequence 1,5,14,30,55,... is

Options

A. 91
B. 85
C. 90
D. 95

Answer

To find the next term in the sequence 1, 5, 14, 30, 55, we can follow these steps:

### Step 1: Identify the sequence
The given sequence is:
1, 5, 14, 30, 55

### Step 2: Calculate the differences between consecutive terms
Let's find the differences between each pair of consecutive terms:
- 5 - 1 = 4
- 14 - 5 = 9
- 30 - 14 = 16
- 55 - 30 = 25

So, the first differences are:
4, 9, 16, 25

### Step 3: Identify the pattern in the first differences
Now, let's look at the first differences:
4, 9, 16, 25

These numbers are:
- 4 = 2²
- 9 = 3²
- 16 = 4²
- 25 = 5²

This shows that the first differences are perfect squares of consecutive integers starting from 2.

### Step 4: Predict the next difference
The next perfect square after 5² (which is 25) is 6²:
- 6² = 36

### Step 5: Add the next difference to the last term of the sequence
Now, we add this next difference to the last term in the original sequence:
- 55 + 36 = 91

### Conclusion
Thus, the next term in the sequence is **91**.

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