Arithmetic sequence questions

Question

1. Consider the arithmetic sequence 34252, 34235, 34218,... Find that last positive term in the sequence.

Since the sequence is decreasing so the formula is

$a_n=-17n+34269$ for finding the nth term

However, I'm not sure how to find the last positive term in the sequence. Should I make $a_n=1$ and solve like this:

$1=-17n+34269$

$-34268=-17n$

$n=2015.76 \approx 2015$

So I would plug in 2015 for n and get 14.

This is how I would think to do it but I'm not sure if I'm correct.

2. Consider again the sequence $2018, 1999, 1980, .... $Find the sum of all the positive terms in the sequence.

Again I have $a_n = -19n+2037$

The sum for arithmetic series is $S_n = \frac{n}{2} (a_1+a_n)$

So $a_1 = 2018$

$a_n$ is the last positive term in the sequence so it is

$1 = -19n+2039$

$n = 107.26 \approx 107$

$a_n = 6$

So $S_n = \frac{107}{2} (2018+6) =108284$

3. Consider the arithmetic sequence 17, 23, 29, 35, ..., 599, 605, whose terms sum to 30789. How many terms are in the sequence?

$30789 = \frac{n}{2} (17+605)$

$n=99$

Answer 1

1. The answer is correct, but I think it's better to use the following reasoning.

We need to find a maximal natural $n$ for which $$-17n+34269>0$$ or $$n<2015.82...,$$ which gives $n=2015$ and $$a_{2015}=14.$$

2. we can solve by the similar way. The answer is true.

   3. Right.