Question

3) Given the sequence-2, 1, 6, 13, 22, ….a) find a closed form formula for anb) find a recursive formula for an.

Answer 1

(a) Notice that:

`\begin{gathered} -2=1^2-3, \\ 1=2^2-3, \\ 6=3^2-3, \\ 13=4^2-3, \\ 22=5^2-3. \end{gathered}`

Therefore a closed-form formula for an is:

`a_n=n^2-3.`

(b) Notice that:

`\begin{gathered} 1=2\cdot2-1+(-2), \\ 6=2\cdot3-1+1, \\ 13=2\cdot4-1+6, \\ 22=2\cdot5-1+13. \end{gathered}`

Therefore a recursive formula for an is:

`\begin{gathered} a_n=2n-1+a_(n-1), \\ a_1=-2. \end{gathered}`

Answer:

(a)

`a_n=n^2-3._{}`

(b)

`\begin{gathered} a_n=2n-1+a_(n-1), \\ a_1=-2. \end{gathered}`