What is the pattern in the sequence 2, 5, 10, 17, 28, 41, 58, 77, 100?

The sequence you provided is 2, 5, 10, 17, 28, 41, 58, 77, 100. To identify the pattern, let's look at the differences between consecutive terms.

Finding the Differences

First, calculate the differences:

• 5 - 2 = 3
• 10 - 5 = 5
• 17 - 10 = 7
• 28 - 17 = 11
• 41 - 28 = 13
• 58 - 41 = 17
• 77 - 58 = 19
• 100 - 77 = 23

Examining the Differences

The differences are: 3, 5, 7, 11, 13, 17, 19, 23. Now, let's look at the differences of these differences:

• 5 - 3 = 2
• 7 - 5 = 2
• 11 - 7 = 4
• 13 - 11 = 2
• 17 - 13 = 4
• 19 - 17 = 2
• 23 - 19 = 4

Identifying the Pattern

The second-level differences alternate between 2 and 4. This suggests that the original sequence is generated by a quadratic function, which typically has a constant second difference.

General Formula

To express the nth term of the sequence, we can use a quadratic formula of the form:

a(n) = An^2 + Bn + C

By solving for A, B, and C using the first few terms, we can derive the specific formula for this sequence.

In summary, the pattern in the sequence involves increasing differences that follow a specific alternating pattern, indicating a quadratic relationship. This means the terms grow at an increasing rate as you progress through the sequence.