sequences and series - Arithmetic progression , find $n$

Given arithmetic progresion $20,17,14,\ldots$

Find smallest value of $n$ so $y_n<0$.

I can find value of $n$ by mind , but do not know how to write the solution.

Answer

Assuming $y_0=20$, your arithmetic progression has closed form

$$y_n=20-3n$$

Now let us solve the following inequality.

$$0>20-3n$$
$$3n>20$$
$$n>\frac{20}{3}=7-\frac{1}{3}$$

The minimum integer value of $n$ that satisfies the condition is

$n=7$.