What's this sequence: 3, 9, 18, 30, 45, 63, 84, 108, 135, 165?

This sequence is: **3, 9, 18, 30, 45, 63, 84, 108, 135, 165**.

To identify its pattern, let's look at the differences between consecutive terms:

- \( 9 - 3 = 6 \)
- \( 18 - 9 = 9 \)
- \( 30 - 18 = 12 \)
- \( 45 - 30 = 15 \)
- \( 63 - 45 = 18 \)
- \( 84 - 63 = 21 \)
- \( 108 - 84 = 24 \)
- \( 135 - 108 = 27 \)
- \( 165 - 135 = 30 \)

We observe that the differences between consecutive terms increase by 3 each time. This suggests that the sequence is generated by adding successive multiples of 3 to each term.

### Formula for the sequence:
The \( n \)-th term of this sequence can be represented by the quadratic equation:

\[
a_n = \frac{3n(n+1)}{2}
\]

This formula reflects a combination of both linear and quadratic growth, where \( a_n \) gives the \( n \)-th term.

Would you like more details or have any questions about this?

### Related Questions:
1. What would be the 11th term in this sequence?
2. How would the sequence look if the difference between terms increased by 5 instead of 3?
3. Can you derive the formula for the \( n \)-th term of this sequence step-by-step?
4. How does this sequence compare to an arithmetic progression?
5. What is the sum of the first 10 terms of this sequence?

### Tip:
Quadratic sequences often arise when the difference between consecutive terms increases by a constant value.

Explore the quadratic sequence 3, 9, 18, 30, 45, 63, 84, 108, 135, 165. Learn how to derive the nth term formula (a_n = 3n(n + 1) / 2) by identifying the increasing differences between consecutive terms. Ideal for Grades 8-10.

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