How to Find the nth Term of a Sequence with a Constant Second Difference (Quadratic Sequence)

Finding the nth term of a sequence with a constant second difference, also known as a quadratic sequence, involves deriving a quadratic expression in the form an² + bn + c. Here's how you can do it:

1. Identify the Sequence as Quadratic:

• Calculate the first difference between consecutive terms (subtract each term from the next term).
• Calculate the second difference (the difference between the first differences).
• If the second difference is constant, the sequence is quadratic.

2. General Formula:

The general formula for the nth term of a quadratic sequence is:

T_n = an² + bn + c

Where:

• T_n is the nth term
• n is the term number
• a, b, and c are constants to be determined

3. Finding the Coefficients a, b, and c:

You can determine the values of a, b, and c by using the following relationships derived from the constant second difference:

• 2a = Second Difference
• 3a + b = First Difference between Term 1 and Term 2
• a + b + c = First Term

4. Steps to Calculate a, b, and c:

1. Calculate 'a': Divide the constant second difference by 2. a = (Second Difference)/2
2. Calculate 'b': Subtract 3a from the first difference between the 1st and 2nd term of the original sequence. b = (First Difference between Term 1 and Term 2) - 3a
3. Calculate 'c': Subtract 'a' and 'b' from the first term of the original sequence. c = (First Term) - a - b

5. Substitute the values into the general formula:

Replace a, b, and c in the formula T_n = an² + bn + c with the values you've calculated.

Example:

Let's consider the sequence: 4, 9, 16, 25, 36...

1. First Difference: 5, 7, 9, 11
2. Second Difference: 2, 2, 2 (Constant, so it's a quadratic sequence)
3. Calculate 'a': a = 2/2 = 1
4. Calculate 'b': The first difference between the first two terms (4 and 9) is 5. b = 5 - (3 * 1) = 2
5. Calculate 'c': The first term is 4. c = 4 - 1 - 2 = 1
6. Substitute: T_n = 1n² + 2n + 1 or T_n = n² + 2n + 1
7. Verification: This sequence is simply (n+1)². Let's check if n=1 yields 4. (1+1)² = 4. This is correct. Similarly, when n=2, (2+1)² = 9. Also correct.

Alternative Method - Using simultaneous equations:

You can also form three equations by substituting n=1, n=2 and n=3 into the general equation T_n = an² + bn + c. Then solve these simultaneous equations to find the values of a, b, and c. This method is longer but can be useful if the relationships in step 3 are difficult to remember.

In Summary:

To find the nth term of a sequence with a constant second difference:

1. Determine that the second difference is constant.
2. Calculate the values of a, b, and c using the formulas 2a = Second Difference, 3a + b = (First Difference between term 1 and term 2), and a + b + c = First Term.
3. Substitute these values into the general quadratic equation T_n = an² + bn + c.
4. Simplify the resulting expression.