To find the [tex]\( n \)[/tex]th term of the given quadratic sequence [tex]\( 3, 8, 15, 24, 35, \ldots \)[/tex], we'll follow a systematic approach. A quadratic sequence generally has the form:
[tex]\[
a n^2 + b n + c
\][/tex]
We need to determine the coefficients [tex]\( a \)[/tex], [tex]\( b \)[/tex], and [tex]\( c \)[/tex]. Here are the steps:
1. Identify the first differences:
- Calculate the differences between consecutive terms of the sequence.
[tex]\[
8 - 3 = 5
\][/tex]
[tex]\[
15 - 8 = 7
\][/tex]
[tex]\[
24 - 15 = 9
\][/tex]
[tex]\[
35 - 24 = 11
\][/tex]
Thus, the first differences are:
[tex]\[
5, 7, 9, 11
\][/tex]
2. Identify the second differences:
- Calculate the differences between consecutive terms of the first differences.
[tex]\[
7 - 5 = 2
\][/tex]
[tex]\[
9 - 7 = 2
\][/tex]
[tex]\[
11 - 9 = 2
\][/tex]
Since the second differences are constant (equal to 2), we confirm that the sequence is quadratic.
3. Form the system of equations:
- Use the first three terms of the sequence to set up equations and solve for [tex]\( a \)[/tex], [tex]\( b \)[/tex], and [tex]\( c \)[/tex].
Let’s use the terms for [tex]\( n = 1, 2, 3 \)[/tex]:
[tex]\[
T_1 = a(1)^2 + b(1) + c = 3 \quad \text{(Equation 1)}
\][/tex]
[tex]\[
T_2 = a(2)^2 + b(2) + c = 8 \quad \text{(Equation 2)}
\][/tex]
[tex]\[
T_3 = a(3)^2 + b(3) + c = 15 \quad \text{(Equation 3)}
\][/tex]
These equations are:
[tex]\[
a + b + c = 3 \quad \text{(Equation 1)}
\][/tex]
[tex]\[
4a + 2b + c = 8 \quad \text{(Equation 2)}
\][/tex]
[tex]\[
9a + 3b + c = 15 \quad \text{(Equation 3)}
\][/tex]
4. Solve the system of equations:
Subtract Equation 1 from Equation 2:
[tex]\[
(4a + 2b + c) - (a + b + c) = 8 - 3
\][/tex]
[tex]\[
3a + b = 5 \quad \text{(Equation 4)}
\][/tex]
Subtract Equation 2 from Equation 3:
[tex]\[
(9a + 3b + c) - (4a + 2b + c) = 15 - 8
\][/tex]
[tex]\[
5a + b = 7 \quad \text{(Equation 5)}
\][/tex]
Subtract Equation 4 from Equation 5:
[tex]\[
(5a + b) - (3a + b) = 7 - 5
\][/tex]
[tex]\[
2a = 2
\][/tex]
[tex]\[
a = 1
\][/tex]
Substitute [tex]\( a = 1 \)[/tex] back into Equation 4:
[tex]\[
3(1) + b = 5
\][/tex]
[tex]\[
3 + b = 5
\][/tex]
[tex]\[
b = 2
\][/tex]
Substitute [tex]\( a = 1 \)[/tex] and [tex]\( b = 2 \)[/tex] into Equation 1:
[tex]\[
1 + 2 + c = 3
\][/tex]
[tex]\[
3 + c = 3
\][/tex]
[tex]\[
c = 0
\][/tex]
5. Write the general formula for the nth term:
Using [tex]\( a = 1 \)[/tex], [tex]\( b = 2 \)[/tex], and [tex]\( c = 0 \)[/tex], we get the nth term as:
[tex]\[
T_n = 1n^2 + 2n + 1 \implies T_n = n^2 + n + 1
\][/tex]
Thus, the nth term of the quadratic sequence [tex]\( 3, 8, 15, 24, 35, \ldots \)[/tex] is:
[tex]\[
n^2 + n + 1
\][/tex]
