Answer:
[tex]2n^2+n+1[/tex]
Step-by-step explanation:
To determine the nth rule for the given sequence, begin by calculating the first differences between consecutive terms by subtracting each term from the next:
[tex]4 \underset{+7}{\longrightarrow} 11 \underset{+11}{\longrightarrow} 22 \underset{+15}{\longrightarrow} 37[/tex]
As the first differences are not the same, we need to calculate the second differences (the differences between the first differences):
[tex]7 \underset{+4}{\longrightarrow} 11 \underset{+4}{\longrightarrow} 15[/tex]
Since the second differences are the same, the sequence is quadratic and will contain an n² term.
The coefficient of the n² term is always half the second difference. As the second difference is 4, the coefficient of the n² term is 2.
Now, compare 2n² with the given sequence (where n is the position of the term in the sequence):
[tex]\begin{array}{|c|c|c|c|c|}\cline{1-5}n&1&2&3&4\\\cline{1-5}2n^2&2&8&18&32\\\cline{1-5}\sf operation&+2&+3&+4&+5\\\cline{1-5}\sf sequence&4&11&22&37\\\cline{1-5}\end{array}[/tex]
We can see that the algebraic operation that takes 2n² to the terms of the sequence is to add one more than the value of n, so to add (n + 1) to 2n².
Therefore, the nth term rule is:
[tex]\Large\boxed{\boxed{2n^2+n+1}}[/tex]