To determine the formula for the nth term of the quadratic sequence 2, 10, 22, 38..., we need to follow these steps:
1. Recognize the form of a quadratic sequence:
A quadratic sequence can be generally represented as:
[tex]\[
T_n = an^2 + bn + c
\][/tex]
where [tex]\( T_n \)[/tex] is the nth term, and [tex]\( a, b, \)[/tex] and [tex]\( c \)[/tex] are constants that we need to determine.
2. Set up equations using known terms of the sequence:
For the given sequence, we can establish equations based on the first few terms.
- When [tex]\( n = 1 \)[/tex]:
[tex]\[
a(1)^2 + b(1) + c = 2 \implies a + b + c = 2
\][/tex]
- When [tex]\( n = 2 \)[/tex]:
[tex]\[
a(2)^2 + b(2) + c = 10 \implies 4a + 2b + c = 10
\][/tex]
- When [tex]\( n = 3 \)[/tex]:
[tex]\[
a(3)^2 + b(3) + c = 22 \implies 9a + 3b + c = 22
\][/tex]
3. Solve the system of equations:
We now have a system of three equations:
[tex]\[
\begin{cases}
a + b + c = 2 \\
4a + 2b + c = 10 \\
9a + 3b + c = 22
\end{cases}
\][/tex]
4. Find the values of [tex]\( a, b, \)[/tex] and [tex]\( c \)[/tex]:
Solving these equations, we get:
[tex]\[
a = 2, \quad b = 2, \quad and \quad c = -2
\][/tex]
5. Formulate the nth term:
Substituting [tex]\( a \)[/tex], [tex]\( b \)[/tex], and [tex]\( c \)[/tex] back into the general form:
[tex]\[
T_n = 2n^2 + 2n - 2
\][/tex]
Thus, the formula for the nth term of the given quadratic sequence is:
[tex]\[
T_n = 2n^2 + 2n - 2
\][/tex]
