How do you write the expression for the $n^{th}$ term of the sequence given $ 0,3,8,15,24 $ ?

Hint: First we will specify all the given terms and then evaluate the values of the required terms from the question. Then we evaluate the value of common difference and solve for the value of $ n $ . We will be using the following formula here: $ a{n^2} \pm bn \pm c $ which is also called structure of a quadratic sequence.

Complete step-by-step answer:
Now in these kinds of questions, we start by first mentioning the given terms first. Also, here if you try to find the difference between the numbers of the sequence, you will notice that they eventually go up by $ 3,5,7 $ and $ 9 $ . Here, these differences have a second difference of $ 2 $ that is they go up by $ 2 $ .
So, to evaluate the first term, here we halve the second difference in order to get the $ a $ coefficient which is $ 1 $ , so that we get,
 $
   = a{n^2} \pm bn \pm c \\
   = {n^2} \pm bn \pm c \;
  $
Now, here if we subtract the original sequence by the sequence of $ {n^2} $ , you will surely notice that there is a common difference which is $ - 1 $ . Hence, there is no additional sequence that you will have to take into consideration.
Hence, the expression for the $n^{th}$ term of the sequence given $ 0,3,8,15,24 $ will be $ {n^2} - 1! $
So, the correct answer is “ $ {n^2} - 1! $ ”.

Note: While solving such types of questions, begin by mentioning all the given terms first in order to avoid any confusion. When you solve for the value of common difference, make sure you substitute the first and second term properly. While applying the quadratic formula, make sure you substitute values of terms along with their signs.