Get personalized answers to your specific questions with IDNLearn.com. Our platform offers comprehensive and accurate responses to help you make informed decisions on any topic.
If the sequence is quadratic, then the n-th term (n ≥ 1) is
[tex]p_n = an^2 + bn + c[/tex]
We're given the first four terms,
[tex]\begin{cases}p_1=-4\\p_2=1\\p_3=12\end{cases}[/tex]
Using the formula for the n-th term, this turns into a system of equations,
[tex]\begin{cases}a+b+c=-4\\4a+2b+c=1\\9a+3b+c=12\end{cases}[/tex]
Solve the system:
• Eliminate c :
(4a + 2b + c) - (a + b + c) = 1 - (-4) ===> 3a + b = 5
(9a + 3b + c) - (a + b + c) = 12 - (-4) ===> 8a + 2b = 16
• Multiply the second equation by 1/2 to get 4a + b = 8, then eliminate b and solve for a :
(4a + b) - (3a + b) = 8 - 5 ===> a = 3
• Solve for b and c :
3a + b = 9 + b = 5 ===> b = -4
a + b + c = 3 - 4 + c = -4 ===> c = -3
Then the rule for the n-th term is
[tex]p_n = \boxed{3n^2 - 4n - 3}[/tex]