How to Find the nth Term of a Quadratic Sequence?

To find the nth term of a quadratic sequence, you can use a general formula and a simple method involving the differences between terms. Here's how:

Understanding Quadratic Sequences

A quadratic sequence is a sequence of numbers where the second differences between consecutive terms are constant. The general formula for a quadratic sequence's nth term is:

T_n = an² + bn + c

Where:

• T_n is the nth term of the sequence.
• n is the position of the term in the sequence (1st, 2nd, 3rd, etc.).
• a, b, and c are constants that need to be determined.

Method to Find a, b, and c

1. Calculate the First Differences: Find the differences between consecutive terms of the sequence.
2. Calculate the Second Differences: Find the differences between the first differences. If these are constant, the sequence is quadratic.
3. Determine 'a': According to the reference, "a is a half of the second difference." In other words:
   a = (Second Difference) / 2
4. Construct a New Sequence: Subtract an² from each term of the original sequence.
5. Find b and c: The new sequence should be a linear sequence, where the difference between each term is now a constant value. Use the method to find the nth term for linear sequences, which is dn + e, where d represents the difference, and e is the zero term (the term before the first term in the linear sequence).
   • b is equal to d and c is equal to e.
6. Write the nth term: Substitute the values of a, b, and c into the general formula: T_n = an² + bn + c

Example

Let's find the nth term of the quadratic sequence: 3, 10, 21, 36...

Sequence (Tn)	3	10	21	36
1st Differences	7	11	15
2nd Differences	4	4

• a: (Second difference)/2 = 4/2 = 2
• an² = 2n².
• Subtract 2n² from the original terms.
  • n = 1 : 3 - (2 * 1²) = 1
  • n = 2 : 10 - (2 * 2²) = 2
  • n = 3 : 21 - (2 * 3²) = 3
  • n = 4 : 36 - (2 * 4²) = 4
• New Sequence is now: 1, 2, 3, 4…

This shows the new sequence has a constant difference of 1 (d) and the zero term (e) is 0 so, the nth term is: 1n + 0, which means b = 1 and c = 0

Therefore, the nth term of the original sequence is :
T_n = 2n² + 1n + 0

Which simplifies to
T_n = 2n² + n

Summary

To find the nth term of a quadratic sequence:

• Identify the first and second differences.
• Calculate 'a' as half the second difference.
• Subtract an² from the original sequence terms.
• Determine the nth term of the resulting linear sequence, which gives 'b' and 'c'.
• Substitute 'a', 'b', and 'c' into the quadratic formula an² + bn + c.