Quadratic Sequences: nth Term

In this video we look at quadratic sequences, and how to find the nth term for them. In quadratic sequences, the first difference changes every time. So instead, we look at the second differences. If the second difference is constant, the sequence is quadratic. This means the nth term rule contains an n-squared term. For the sequence 5, 9, 15, 23, 33 start by finding the first and second differences. The second difference is constant at 2. Because there is a constant second difference so it’s going to be n-squared. As the second difference is 2, it will be 1 n-squared. Always half the second difference. Now write out the original sequence: 5, 9, 15, 23, 33. And the 1n-squared underneath it: 1, 4, 9, 16, 25. Compare the difference. So 5 to 1 is 4. 8 to 4 is 4. 13 to 9 is 4, and so on. All you need to do is find the nth term of this sequence, and you have your quadratic nth term formula: n-squared + 4. For this sequence: 5, 12, 23, 38, 57, ... the second difference is 4 and so the quadratic will be 2n-squared as we have halved the second difference. Now write out the original sequence: 5, 12, 23, 38, 57 and 2 n-squared underneath it: 2, 8, 18, 32, 50. Find the difference between the two: 3, 4, 5, 6, 7. The nth term rule for this is n + 2. So the nth term rule for this quadratic formula is 2n2 + n + 2.