Question

What is the formula for the nth term of a quadratic sequence?

Answer 1

A quadratic sequence is a sequence of numbers in which each term is generated by a quadratic function. The formula for the nth term of a quadratic sequence is:

\[a_n = a_1 + (n-1)d + \frac{(n-1)(n-2)r}{2}\]

Where:

• \(a_1\) is the first term of the sequence
• \(d\) is the common difference between consecutive terms
• \(r\) is the common ratio between consecutive terms

For example, if the first term of a quadratic sequence is 2, the common difference is 3 and the common ratio is 4, the quadratic sequence is: \(2,5,14,30,62,126...\)

The nth term of this sequence can be found by plugging the values into the formula: \[ a_n = 2 + (n-1)3 + \frac{(n-1)(n-2)4}{2}\]

So, the fifth term of this sequence is 30, the 8th term is 126 and so on.

It's also possible to write the nth term of a quadratic sequence using a quadratic function, for example: \[a_n = a + (n-1)b + \frac{(n-1)(n-2)c}{2}\] where \(a, b, c\) are the constant coefficients of the quadratic function.