How to Find the nth Term with a Second Difference?

To find the nth term of a sequence with a constant second difference, follow these steps, incorporating the techniques outlined in the provided reference:

Understanding Sequences with Second Differences

Sequences where the second difference between terms is constant are quadratic sequences. The general form of the nth term of a quadratic sequence is:

an² + bn + c

Where 'a', 'b', and 'c' are constants. The method below allows you to determine these constants and thus find the nth term.

Steps to Finding the nth Term

Here’s a breakdown of the process, based on the reference information, with examples:

1. Calculate the Second Difference:

   • Find the difference between consecutive terms in the sequence. This is the first difference.
   • Find the difference between consecutive terms in the first difference. This is the second difference.
   • Confirm that the second difference is constant. If it isn't, this method won't work.

   Example: Consider the sequence: 7, 12, 19, 28, 39...

   • First difference: 5, 7, 9, 11...
   • Second difference: 2, 2, 2... (Constant, so we can proceed)

2. Determine 'a':

   • The constant second difference is equal to 2a. Therefore, a = (second difference) / 2.

   Example (continued):

   • Second difference = 2
   • a = 2 / 2 = 1

   So, our nth term formula is beginning to look like: 1n² + bn + c

3. Subtract an² from the Original Sequence:

   • Create a new sequence by subtracting an<sup>2</sup> (which is n<sup>2</sup>) from each term of the original sequence.

   Example (continued):

   n	Original Term	an2 (n2)	Original Term - an2
   1	7	1	6
   2	12	4	8
   3	19	9	10
   4	28	16	12
   5	39	25	14

   The new sequence is: 6, 8, 10, 12, 14...

4. Find the nth Term of the Arithmetic Sequence:

   • The sequence resulting from the subtraction in step 3 will be an arithmetic sequence (constant first difference). Find the nth term of this arithmetic sequence. This will be in the form bn + c.

   Example (continued):

   • The arithmetic sequence is: 6, 8, 10, 12, 14...
   • The difference between terms is 2. So b = 2
   • When n = 1, the term is 6. So, 2(1) + c = 6. Therefore, c = 4.
   • The nth term of the arithmetic sequence is: 2n + 4

5. Combine the Results:

   • Combine the an<sup>2</sup> term with the nth term of the arithmetic sequence to get the nth term of the original quadratic sequence.

   Example (continued):

   • an² = 1n² = n²
   • nth term of arithmetic sequence = 2n + 4
   • Therefore, the nth term of the original sequence is: n² + 2n + 4

Summary Table

Step	Description	Example (Using 7, 12, 19, 28, 39...)
1. Calculate Second Difference	Find the constant difference between the first differences.	Second difference = 2
2. Determine 'a'	a = (Second Difference) / 2	a = 2 / 2 = 1
3. Subtract an2	Subtract an2 from the original sequence.	New Sequence: 6, 8, 10, 12, 14...
4. Find nth Term of Arithmetic Sequence	Determine the nth term of the new arithmetic sequence resulting from step 3.	2n + 4
5. Combine Results	Combine an<sup>2</sup> and the arithmetic nth term.	n2 + 2n + 4