Verify whether the following numbers belong to every sequence, and in affirmative case state what position they occupy:

a) $$25$$. Does it belong to the sequence $$a_n=(3n+4)_{n\in\mathbb{N}}$$?

b) $$\dfrac{9}{5}$$. Does it belong to the sequence $$c_n=\dfrac{n^2}{n+1}$$?

### Development:

a) The number $$25$$ will belong to the sequence if there exists a natural number $$n$$ in such a way that $$a_n=25$$. Since $$a_n=3n+4$$, replacing an equality in the other one we have: $$$25=3n+4$$$ $$$3n=21$$$ $$$n=7$$$

So, not only do we see that it belongs to the sequence, but we have figured out which position it occupies.

b) We will proceed as in the previous case: $$$\left. \begin{array}{c} c_n=\dfrac{9}{5} \\ c_n=\dfrac{n^2}{n+1} \end{array} \right\} \Rightarrow \dfrac{9}{5}=\dfrac{n^2}{n+1} \Rightarrow 9(n+1)=5n^2$$$

So we can only solve the equation of the second grade:

$$$5n^2-9n-1=0 \Rightarrow n=\dfrac{9\pm3\sqrt{5}}{10}$$$ Since we have not obtaind any natural value for $$n$$, the number $$\dfrac{9}{5}$$ is not part of the sequence (it cannot occupy an irrational position!).

### Solution:

a) $$a_7=25$$

b) It does not belong to the sequence.