Question

Consider the sequence below:-2,1,6,13,22, ....What explicit expression can be used to find the nth term of this sequence?

Answer 1

f(n)=n²-3

Step-by-step explanation:

In the sequence:

`-2,1,6,13,22,...`

First, we find the difference between the terms.

`\begin{gathered} 1-(-2)=3 \\ 6-1=5 \\ 13-6=7 \\ 22-13=9 \end{gathered}`

It is observed that the difference between successive terms is the addition of consecutive odd numbers.

This is an example of a quadratic sequence.

The general form of a quadratic sequence is:

`\begin{gathered} f(n)=an^2+bn+c \\ f(1)=-2 \\ \implies a+b+c=-2 \\ f(2)=1 \\ \implies4a+2b+c=1 \\ f(3)=6 \\ \implies9a+3b+c=6 \end{gathered}`

If we solve the system of equations:

`\begin{gathered} a+b+c=-2 \\ 4a+2b+c=1 \\ 9a+3b+c=6 \\ a=1,b=0,c=-3 \end{gathered}`

The explicit expression for this sequence is:

`f(n)=n^2-3`