Say you get 20 dollars in January, and 30 in February, and 40 in March, and so on. You can model that as receiving $10x+10$ dollars every month.
Your total received money would be 20 dollars, then 50, then 90, and so on. How do you calculate this total for any future month? And how would you go about finding the function of that total?
Integration is the first thing I thought of, but it assumes a continuous gain, not a discrete gain.
You can sum up the sequence.
$$S_n=\sum_{k=1}^{n} 10\cdot k+10=\sum_{k=1}^{n} 10\cdot k+\sum_{k=1}^{n} 10$$
$$=10 \cdot \sum_{k=1}^{n} k+10n=10\cdot \frac{n\cdot (n+1)}{2}+10n$$
The closed formula for the sum is well known, see here. A little more simplification.
$S_n=10\cdot \frac{n\cdot (n+3)}{2}$
For instance $S_3=10\cdot \frac{3\cdot 6}{2}=90$
This is a arithmetic sequence with common difference $d=10$ and initial term $a=20$ the sum of the first $n$ can be given by $$S_n=\frac{n}2(2a+(n-1)d)=\frac{n}2(40+(n-1)\cdot 10)=5n^2+15n$$
As the growth is linear, the average amount is also the average of the first and last amounts. So the total is the average of the first and last amounts, times the number of months.
E.g.
$$30+40+50=3\,\frac{30+50}2,$$
$$30+40+50+60+70+80+90=7\,\frac{30+90}2.$$
