I often run into the following situation:

- In January, I'm producing $10$ items/month.
- By December, I want to be producing $65$ items/month.
- So assuming things improve linearly, my production is increasing by $5$ items/month.
- Question: How many total items do I produce during the year?

I could be wrong, but I think this can be represented like this:

$$\sum_{n=1}^N{a+\frac{n \cdot (z - a)}{N-1}}=\sum_{n=1}^{12}{10+\frac{n \cdot (65 - 10)}{12-1}}=\sum_{n=1}^{12}{10+5n}$$

I want to be able to roughly estimate the answer in my head for any given start/target numbers $a$ and $z$. How can I do that?

P.S. I have no background in math, so apologies for awkward wording, incorrect terminology, etc.