Let's say I'm spending $\$10$ a month on a service today, and the cost grows by $10\%$ a month. How much have I spent in a year?
I know that in $12$ months, I'll be spending $\$10\times 1.1^{12}$ per month, but I am not sure how to figure out the total spent along the way without adding all the individual monthly amounts as a large manual calculation.
@John Omielan has a perfect hint, try using a geometric series.
The linked resource in John's comment has the formula for a geometric series, which is $$\sum_{k=0}^{n-1}ar^{k}= a(\frac{1-r^{n}}{1-r})$$ This represents the sum from time $0$ to time $n=12$ to represent twelve months in a year at a rate $r=1.10$ to represent the $10$% increase per month. $a$ is your $\$10$ per month.
Now, although it is not mentioned in your question, let us assume we are making a payment at the beginning of the month as you mentioned the service starts today, so the first payment is at time $0$ for $\$10$ and the $12$th and final payment is at time $11$ for $\$10*(1.10)^{11}$. This will be assuming $12$ total payments in the one year period.
Plug these values into your equation and you get $$\$10(\frac{1-(1.10)^{12}}{1-(1.10)})$$
Try evaluating and this should give you your answer. I hope this helps and welcome to Math SE!