Answer:
[tex]n^2+5n+2[/tex]
Step-by-step explanation:
Given the sequence
8, 16, 26, 38, 52, 68, 86,...
The nth term of a quadratic sequence will take the form:
[tex]an^2+bn+c[/tex]
Step 1: Find the difference between each term
The differences are: 8,10,12,14,...
Step 2: Find the differences between the differences
We notice an addition of 2, so the second difference is 2.
Step 3:
Take the half of the second difference to obtain a.
2/2=1
Therefore: a=1
The sequence for now is: [tex]n^2+bn+c[/tex]
Step 4: Find [tex]T_n-n^2[/tex]
[tex]T_1=8, T_1-1^2=8-1=7\\T_2=16, T_2-2^2=16-4=12\\T_3=26, T_3-3^2=26-9=17[/tex]
We notice that [tex]T_n-n^2[/tex] forms a linear sequence 7,12,17
We can write this as 2+5n
Therefore, Our nth rule for the quadratic sequence is therefore: [tex]n^2+5n+2[/tex]
