Find the nth term of the sequence

9, 24, 45, 72, 105, …

The given sequence is:  
\[ 9, 24, 45, 72, 105, \dots \]

To find the general formula for the nth term of this sequence, let's analyze the pattern.

1. **First differences**:  
\[ 24 - 9 = 15 \]  
\[ 45 - 24 = 21 \]  
\[ 72 - 45 = 27 \]  
\[ 105 - 72 = 33 \]

The first differences are increasing by 6 each time, indicating a quadratic sequence.

2. **Second differences**:  
\[ 21 - 15 = 6 \]  
\[ 27 - 21 = 6 \]  
\[ 33 - 27 = 6 \]

Since the second differences are constant, this is a quadratic sequence. The general form of the nth term for a quadratic sequence is:

\[
T_n = an^2 + bn + c
\]

We now need to determine the coefficients \( a \), \( b \), and \( c \).

3. **Setting up equations**:  
We know the first few terms of the sequence:
- \( T_1 = 9 \)
- \( T_2 = 24 \)
- \( T_3 = 45 \)

Substitute these into the general form:

\[
T_1 = a(1)^2 + b(1) + c = 9 \quad \Rightarrow \quad a + b + c = 9
\]
\[
T_2 = a(2)^2 + b(2) + c = 24 \quad \Rightarrow \quad 4a + 2b + c = 24
\]
\[
T_3 = a(3)^2 + b(3) + c = 45 \quad \Rightarrow \quad 9a + 3b + c = 45
\]

We now have a system of three equations:
1. \( a + b + c = 9 \)
2. \( 4a + 2b + c = 24 \)
3. \( 9a

Find the nth term of the sequence 9, 24, 45, 72, 105, …

Discover how to find the nth term of the quadratic sequence 9, 24, 45, 72, 105 with a step-by-step solution. Learn about quadratic sequences, constant second differences, and the general formula T_n = an^2 + bn + c. Suitable for students in Grades 9-11.

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