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.
This is known as arithmetic series.
If we start from $a$ and ends with $z$ in $N$ period, the formula is just
$$\frac{N}{2}(a+z).$$
The idea is to pair up the $i$-th number with the $(N-i)$-th number and we notice that the sum is equal for all $i$.