Question

How do you express the sequence below as a recursively defined function 4, 11, 25, 53, 109,...?

Answer 1

Sequence as a recursively defined function is

`a_n=a_(n-1)+7(n-1)`, where `a_1=4`

Explanation:

Generally the functions are defined explicitly by a formula in terms of the variable, say `x`.

We can also define functions recursively: in terms of the same function of a smaller variable building on itself and in the case of sequences, this is generally the number of the term.

For example here, first term is `a_1=4`.

Observe that second term `a_2=11` is obtained by adding `7` to first term

and third term `a_3=25` obtained by adding `14` to second term.

Hence we can say `a_1=4`

`a_2=4+7(2-1)=11`

and `a_3=a_2+7(3-1)=11+7xx2=21`

Hence we can say `a_n=a_(n-1)+7(n-1)`, where `a_1=4`