The given sequence is:
$3, 6, 11, 18, 27, 38, 51$

To find the next term in this quadratic sequence, we can start by examining the differences between consecutive terms:

1. $6 - 3 = 3$
2. $11 - 6 = 5$
3. $18 - 11 = 7$
4. $27 - 18 = 9$
5. $38 - 27 = 11$
6. $51 - 38 = 13$

The first differences are: $3, 5, 7, 9, 11, 13$

Next, let's examine the second differences, which are the differences between the first differences:

1. $5 - 3 = 2$
2. $7 - 5 = 2$
3. $9 - 7 = 2$
4. $11 - 9 = 2$
5. $13 - 11 = 2$

Since the second differences are constant ($2$), this confirms that the sequence is quadratic.

To find the next term, we extend the first differences by adding 2 to the last first difference: $13 + 2 = 15$

Now, add this new first difference to the last term of the sequence: $51 + 15 = 66$

Thus, the next term in the sequence is 66.

Would you like further details or clarifications? Here are five related questions:

1. How do you find a general formula for a quadratic sequence?
2. Can quadratic sequences be used to model real-world situations?
3. What is the difference between arithmetic, geometric, and quadratic sequences?
4. How can we determine the nth term of this quadratic sequence?
5. What happens to the sequence if we change the second difference?

Tip: Always check the second differences for quadratic sequences—constant second differences confirm a quadratic pattern!