Answer:
The nth term of the quadratic sequence is [tex]n^2+2n[/tex].
Step-by-step explanation:
Since the given sequence is a quadratic sequence.
Let the terms be of the form [tex]an^2+bn+c[/tex].
So, we have,
At n= 1, [tex]a+b+c=3[/tex] .......................(1)
At n= 2, [tex]4a+2b+c=8[/tex] .......................(2)
At n=3, [tex]9a+3b+c=15[/tex] .......................(3)
By (1) and (2), we have,
[tex]3-a-b=8-4a-2b[/tex] i.e. [tex]3a+b=5[/tex] ...............(4)
By (1) and (3), we have,
[tex]3-a-b=15-9a-3b[/tex] i.e. [tex]8a+2b=12[/tex] i.e. [tex]4a+b=6[/tex] ........(5)
Subtracting (4) and (5), we get, [tex]a=1[/tex]
So, (4) implies [tex]3\times 1+b=5[/tex] i.e. [tex]b= 5-3[/tex] i.e. [tex]b= 2[/tex]
So, from equation (1), we get,
[tex]a+b+c=3[/tex] i.e. [tex]1+2+c=3[/tex] i.e. [tex]3+c=3[/tex] i.e. [tex]c=0[/tex]
Hence, the terms of the given sequence are of the form, [tex]n^2+2n[/tex]
That is, the nth term of the quadratic sequence is [tex]n^2+2n[/tex].
