To find the $n$th term of the sequence:
3, 12, 27, 48, ...

Step 1: Identify the Pattern

Let's check the differences between consecutive terms:

• $12 - 3 = 9$
• $27 - 12 = 15$
• $48 - 27 = 21$

The first differences are: 9, 15, 21, ...
Checking second differences:

• $15 - 9 = 6$
• $21 - 15 = 6$

Since the second difference is constant (6), the sequence follows a quadratic pattern of the form:

$a_n = An^2 + Bn + C$

Step 2: Solve for A, B, and C

Using given terms:

1. $a_1 = 3 \Rightarrow A(1)^2 + B(1) + C = 3 \Rightarrow A + B + C = 3$
2. $a_2 = 12 \Rightarrow A(2)^2 + B(2) + C = 12 \Rightarrow 4A + 2B + C = 12$
3. $a_3 = 27 \Rightarrow A(3)^2 + B(3) + C = 27 \Rightarrow 9A + 3B + C = 27$

Step 3: Solve the System

$A + B + C = 3$ $4A + 2B + C = 12$ $9A + 3B + C = 27$

Subtract the first equation from the second:

$(4A + 2B + C) - (A + B + C) = 12 - 3$

$3A + B = 9$

Subtract the second from the third:

$(9A + 3B + C) - (4A + 2B + C) = 27 - 12$

$5A + B = 15$

Now, solve:

1. $3A + B = 9$
2. $5A + B = 15$

Subtract:

$(5A + B) - (3A + B) = 15 - 9$

$2A = 6 \Rightarrow A = 3$

Substituting $A = 3$ into $3A + B = 9$:

$3(3) + B = 9 \Rightarrow 9 + B = 9 \Rightarrow B = 0$

Substituting $A = 3$, $B = 0$ into $A + B + C = 3$:

$3 + 0 + C = 3 \Rightarrow C = 0$

Step 4: Final Formula

$a_n = 3n^2$

Thus, the nth term of the sequence is:

$a_n = 3n^2$

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Do you need more details or have any questions?

Here are 5 related questions to explore further:

1. What is the 10th term of this sequence?
2. How do you determine if a sequence follows a quadratic pattern?
3. Can you derive a formula for a cubic sequence?
4. What is the sum of the first 5 terms of this sequence?
5. How can you generalize the method for any polynomial sequence?

Tip: Always check second differences for quadratic sequences!