Answer:
The next term is 44.
Step-by-step explanation:
From the statement of the problem we know that the n-th term is given by a quadratic expression. So, if we denote the n-th term by [tex]a_n[/tex], then it can be written as [tex]a_n = an^2+bn+c.[/tex] Then, we have to find the coefficients [tex]a[/tex], [tex]b[/tex] and [tex]c[/tex].
The idea here is to obtain a system of linear equations with [tex]a[/tex], [tex]b[/tex] and [tex]c[/tex] as unknowns. As we need three of those equation we substitute [tex]n=1[/tex], [tex]n=2[/tex] and [tex]n=3[/tex] in the expression for [tex]a_n[/tex]. Thus
[tex]-7=a+b+c[/tex]
[tex]-6=a2^2+b2+c = 4a+2b+c[/tex]
[tex]-3=a3^2+b3+c = 9a+3b+c.[/tex]
The solutions of this system of linear equations is [tex]c=-6[/tex], [tex]b=-2[/tex] and [tex]a=1[/tex].
Hence, [tex]a_n = n^2-2n-6.[/tex]
Notice that,
[tex]a_4 = 4^2-2\cdot 4-6 = 16-8-6=2,[/tex]
[tex]a_5 = 5^2-2\cdot 5-6 = 9.[/tex]
The above values correspond with ones given in the statement of the exercise. As we have 7 values given, we need to find the 8th:
[tex]a_8 = 8^2-2\cdot 8-6 = 64-16-6=64-22 = 44,[/tex]
