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

Feb 11, 2017

Sequence as a recursively defined function is

${a}_{n} = {a}_{n - 1} + 7 \left(n - 1\right)$, 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 ${a}_{1} = 4$

${a}_{2} = 4 + 7 \left(2 - 1\right) = 11$

and ${a}_{3} = {a}_{2} + 7 \left(3 - 1\right) = 11 + 7 \times 2 = 21$

Hence we can say ${a}_{n} = {a}_{n - 1} + 7 \left(n - 1\right)$, where ${a}_{1} = 4$