For those interested: We are dealing here with a recurring relationship, or a recursive equation. I calculate per week, since 950 euros are added every day, this is 6650 euros per week, and the previous amount is doubled in this week.
So we get the relation A_(n+1) = 2*A_n+6650, where A_n is the fine in week n. After all, next week’s amount (A_(n+1) is double this week’s fine (so 2*A_n) and an additional 6650 euros.
To solve a recurring relationship you also need a starting value, an A_0. We assume that the fine starts at 0 euros, so A_0=0. Now we see
A_1=6650
A_2 = 2(6650)+6650 =3*6650.
A_3 = 2(3*6650)+6650 =7*6650.
A_4 = 2(7*6650)+6650 =15*6650.
A little tweaker will quickly see the powers of 2 minus 1 in the pattern of 1,3,7,15. Indeed, working out a direct function for this recursive equation provides a solution
A_n = A_0*2^n + (2^n -1)*6650, and since A_0 was equal to 0 this term is canceled and it is (2^n -1)*6650
If we now take A_0 = 1.905978*10^34, so we start with this week with the current fine amount as the starting value, we indeed quickly see that the term (2^n -1)*6650 now no longer contributes anything to the total. The exact solution is 218.3167, which means that after 218 weeks there is still no Googol fine. We will have to wait until the 219th doubling, with which the fine will be 1.60578*10^100 euros
A Deeper Look into Recursive Fines: Up, Up and Away!
Ah, recursive equations! The mathematical equivalent of those excruciatingly long phone queues where you’re twice as annoyed every minute, yet somehow it’s always just one more minute. Today, we’re diving into a fine that compounds so rapidly it feels like someone’s launched it into a black hole. Buckle up, because we’re going on a journey of euros, powers, and bewilderingly large numbers!
How Do We Get to 6650 Euros per Week?
Now, grab your calculators, folks! We’ll be racking up 950 euros every single day, which turns into a staggering 6650 euros each week. That’s right; it’s not just a fine—it’s a full-blown inflation party! And, in case anyone thought they could just ignore it, next week’s amount doubles the previous amount and adds a little something-something on top: an extra 6650 euros. Yes, it’s that friend who keeps on bringing snacks, only for you to realize they’re expensive delicacies!
The Equation That Has Us All in a Fuddle
Here’s the juicy mathematical bit: A_(n+1) = 2*A_n + 6650. Think of it like a game of Monopoly where every time you pass GO, you just get more and more money… except you owe it to the bank and they refuse to let you purchase Boardwalk. To truly navigate this fine mess, we need a starting value, or as I like to call it, our “fine foundation.” In this case, we start with nothing: A_0 = 0.
What Happens as Weeks Go By?
So, what do we get when we plug in some numbers? Week one gives us A_1 = 6650. Week two? A_2 = 3*6650. By week three, we’re at A_3 = 7*6650. By week four, it’s A_4 = 15*6650. If this were a sitcom, it’d be titled The Fines Keep Coming!.
Ah, The Power of Two
Notice a pattern? That delightful string of numbers: 1, 3, 7, 15… it feels like I’m playing the world’s most boring game of darts! But wait, there’s magic in that madness. Every number is a power of 2 minus 1 (thank you, ancient mathematicians!). With some tweaking, we find the direct formula: A_n = A_0*2^n + (2^n - 1)*6650. And if you cancelled out our starting value of 0, it boils down to (2^n - 1)*6650. Easy peasy, like counting sheep on a plane!
The Absurdity at Work
Now let’s spice things up a bit! Imagine you start off with a sumptuous amount: A_0 = 1.905978*10^34. Sounds like a wallet for the richest of the rich! At this point, you may think, “Pfft! What’s a few billions in fines?” But I’m afraid even that hefty sum will be dwarfed by the escalating fines. After 218 weeks of penance, your final tally would be about 218.3167, which is practically chicken feed considering that by the 219th week, we’re looking at a mind-boggling, jaw-dropping 1.60578*10^100 euros!.
Conclusion: A Fine Exponential Journey
So there you have it, folks! Mathematics really is the gift that keeps on giving… just not in the way you’d like! You see, when it comes to our fines, they might just keep doubling until the end of time, or until you finally manage to pay them off, whichever comes first. So, if you find yourself faced with a recursive fine equation, just remember: it isn’t just numbers; it’s an entertaining little game of financial doom wrapped in witty math. Thanks for joining me in this mathematical joyride!
For those interested in mathematical relationships: We are examining a recursive equation that defines a pattern where a specific amount is consistently added over time. Specifically, I calculate weekly totals based on the addition of 950 euros each day. Over the span of a week, this accumulates to a total of 6,650 euros, while concurrently, the previous week’s total is effectively doubled.
This results in the relationship A_(n+1) = 2*A_n + 6650, where A_n represents the fine accumulated in week n. Importantly, the amount for the following week, A_(n+1), is derived from simply doubling the current week’s fine (2*A_n) and adding the additional 6,650 euros.
To effectively resolve this recurring relationship, we require a starting value designated as A_0. For our purposes, we assume the fine begins at 0 euros, leading us to set A_0 = 0. Thus, we can observe the computations unfold as follows:
A_1 = 6650
A_2 = 2(6650) + 6650 = 3 * 6650.
A_3 = 2(3 * 6650) + 6650 = 7 * 6650.
A_4 = 2(7 * 6650) + 6650 = 15 * 6650.
A close examination of this sequence reveals the powers of 2 minus 1 reflected in the resulting patterns of 1, 3, 7, and 15. In fact, isolating a direct function that encapsulates this recursive equation leads us to the formula A_n = A_0 * 2^n + (2^n – 1) * 6650. Given that A_0 is set to 0, this effectively cancels out the first term, simplifying our expression to (2^n – 1) * 6650.
Now, let’s establish A_0 = 1.905978 * 10^34. In this scenario, we initiate our calculations with the current fine amount as the base value. Here, we promptly find that the term (2^n – 1) * 6650 no longer plays a significant role in the overall total. The precise outcome after 218 weeks surfaces as 218.3167, indicating that the Googol fine remains elusive. We will inevitably have to anticipate the 219th doubling, at which point the fine will escalate astronomically to 1.60578 * 10^100 euros.
