Answer :
To find the [tex]\( n^{\text{th}} \)[/tex] term rule for the given quadratic sequence [tex]\( 8, \, 11, \, 16, \, 23, \, 32, \, \ldots \)[/tex], follow these steps:
1. Identify the First Differences:
[tex]\[ \begin{align*} \Delta_1 & = 11 - 8 = 3 \\ \Delta_2 & = 16 - 11 = 5 \\ \Delta_3 & = 23 - 16 = 7 \\ \Delta_4 & = 32 - 23 = 9 \\ \end{align*} \][/tex]
These are the first differences of the sequence.
2. Identify the Second Differences:
[tex]\[ \begin{align*} \Delta^2_1 & = 5 - 3 = 2 \\ \Delta^2_2 & = 7 - 5 = 2 \\ \Delta^2_3 & = 9 - 7 = 2 \\ \end{align*} \][/tex]
Since the second differences are constant (all equal to 2), the sequence is quadratic.
3. Assume the General Form of the Quadratic Sequence:
The general form of the [tex]\( n^{\text{th}} \)[/tex] term of a quadratic sequence is:
[tex]\[ a n^2 + b n + c \][/tex]
4. Set Up Equations Using Known Terms:
Substitute the first few terms to find [tex]\( a \)[/tex], [tex]\( b \)[/tex], and [tex]\( c \)[/tex]:
[tex]\[ \begin{align*} \text{For } n = 1: & \quad a(1)^2 + b(1) + c = 8 \\ \text{For } n = 2: & \quad a(2)^2 + b(2) + c = 11 \\ \text{For } n = 3: & \quad a(3)^2 + b(3) + c = 16 \\ \end{align*} \][/tex]
This results in the following system of equations:
[tex]\[ \begin{align*} a + b + c &= 8 \quad \text{(i)} \\ 4a + 2b + c &= 11 \quad \text{(ii)} \\ 9a + 3b + c &= 16 \quad \text{(iii)} \\ \end{align*} \][/tex]
5. Solve the System of Equations:
Subtract equation (i) from equation (ii) to eliminate [tex]\( c \)[/tex]:
[tex]\[ (4a + 2b + c) - (a + b + c) = 11 - 8 \\ 3a + b = 3 \quad \text{(iv)} \][/tex]
Similarly, subtract equation (ii) from equation (iii):
[tex]\[ (9a + 3b + c) - (4a + 2b + c) = 16 - 11 \\ 5a + b = 5 \quad \text{(v)} \][/tex]
Subtract equation (iv) from equation (v):
[tex]\[ (5a + b) - (3a + b) = 5 - 3 \\ 2a = 2 \\ a = 1 \][/tex]
Substitute [tex]\( a = 1 \)[/tex] into equation (iv):
[tex]\[ 3(1) + b = 3 \\ 3 + b = 3 \\ b = 0 \][/tex]
Substitute [tex]\( a = 1 \)[/tex] and [tex]\( b = 0 \)[/tex] into equation (i):
[tex]\[ 1 + 0 + c = 8 \\ c = 7 \][/tex]
6. Write the Final [tex]\( n^{\text{th}} \)[/tex] Term Rule:
So, the [tex]\( n^{\text{th}} \)[/tex] term of the sequence is:
[tex]\[ n^2 + 7 \][/tex]
Therefore, the [tex]\( n^{\text{th}} \)[/tex] term rule for the given quadratic sequence is:
[tex]\[ n^2 + 7. \][/tex]
1. Identify the First Differences:
[tex]\[ \begin{align*} \Delta_1 & = 11 - 8 = 3 \\ \Delta_2 & = 16 - 11 = 5 \\ \Delta_3 & = 23 - 16 = 7 \\ \Delta_4 & = 32 - 23 = 9 \\ \end{align*} \][/tex]
These are the first differences of the sequence.
2. Identify the Second Differences:
[tex]\[ \begin{align*} \Delta^2_1 & = 5 - 3 = 2 \\ \Delta^2_2 & = 7 - 5 = 2 \\ \Delta^2_3 & = 9 - 7 = 2 \\ \end{align*} \][/tex]
Since the second differences are constant (all equal to 2), the sequence is quadratic.
3. Assume the General Form of the Quadratic Sequence:
The general form of the [tex]\( n^{\text{th}} \)[/tex] term of a quadratic sequence is:
[tex]\[ a n^2 + b n + c \][/tex]
4. Set Up Equations Using Known Terms:
Substitute the first few terms to find [tex]\( a \)[/tex], [tex]\( b \)[/tex], and [tex]\( c \)[/tex]:
[tex]\[ \begin{align*} \text{For } n = 1: & \quad a(1)^2 + b(1) + c = 8 \\ \text{For } n = 2: & \quad a(2)^2 + b(2) + c = 11 \\ \text{For } n = 3: & \quad a(3)^2 + b(3) + c = 16 \\ \end{align*} \][/tex]
This results in the following system of equations:
[tex]\[ \begin{align*} a + b + c &= 8 \quad \text{(i)} \\ 4a + 2b + c &= 11 \quad \text{(ii)} \\ 9a + 3b + c &= 16 \quad \text{(iii)} \\ \end{align*} \][/tex]
5. Solve the System of Equations:
Subtract equation (i) from equation (ii) to eliminate [tex]\( c \)[/tex]:
[tex]\[ (4a + 2b + c) - (a + b + c) = 11 - 8 \\ 3a + b = 3 \quad \text{(iv)} \][/tex]
Similarly, subtract equation (ii) from equation (iii):
[tex]\[ (9a + 3b + c) - (4a + 2b + c) = 16 - 11 \\ 5a + b = 5 \quad \text{(v)} \][/tex]
Subtract equation (iv) from equation (v):
[tex]\[ (5a + b) - (3a + b) = 5 - 3 \\ 2a = 2 \\ a = 1 \][/tex]
Substitute [tex]\( a = 1 \)[/tex] into equation (iv):
[tex]\[ 3(1) + b = 3 \\ 3 + b = 3 \\ b = 0 \][/tex]
Substitute [tex]\( a = 1 \)[/tex] and [tex]\( b = 0 \)[/tex] into equation (i):
[tex]\[ 1 + 0 + c = 8 \\ c = 7 \][/tex]
6. Write the Final [tex]\( n^{\text{th}} \)[/tex] Term Rule:
So, the [tex]\( n^{\text{th}} \)[/tex] term of the sequence is:
[tex]\[ n^2 + 7 \][/tex]
Therefore, the [tex]\( n^{\text{th}} \)[/tex] term rule for the given quadratic sequence is:
[tex]\[ n^2 + 7. \][/tex]
