How to solve this quick pre-algebra question?
Logan the Leprechaun loves Lucky Lollipops. He decides to increase his luck by sharing it. First, Logan will eat one Lucky Lollipop each day. Second, Logan will give away 1 Lucky Lollipops on the first day and, for the ensuing days, will give away twice as many Lucky Lollipops as he gave on way the previous day plus one more.How many Lucky Lollipops will Logan give away on the 12th day, and the 24th day? Supposed Logan, so enjoying all the extra luck, decides to continue this fashion indefinitely. How many Lucky Lollipops would Logan give away on the nth day?
What would be the general equation that describes how much lucky lollipop logan would give away on the nth day?
Pls help ASAP :') Thank you so much XD
It'd be great if you can explain this problem step by step :D
- SamwiseLv 78 years agoFavorite Answer
The first day, he gives away 1 lollipop.
The second day, he gives away 2(1) + 1 = 3 lollipops.
The third day, he gives away 2(3) + 1 = 7 lollipops.
The fourth day, he gives away 2(7) + 1 = 15 lollipops.
The fifth day, he gives away 2(15) + 1 = 31 lollipops.
I hope the pattern is becoming clear:
The sequence is 1, 3, 7, 15, 31 ...
If not, here's a hint: the powers of two are 2, 4, 8, 16, 32 ...
So we suspect that on day n, he gives away 2^n - 1 lollipops.
[I'm using ^ as an exponent indicator.]
We can see that's true for the first five days, but is it true in general?
Well, if it's true for some particular day k, then
on day k he gives away 2^k - 1 lollipops
and on day (k+1) he gives away
2 (2^k - 1) + 1 = 2 (2^k) - 2 + 1 = 2^(k+1) - 1 lollipops.
So if the pattern holds for any day (and we already KNOW it holds for days 1 through 5),
it holds for the next day, and then it must hold for the day after that, and so on...
It holds for any possible value of n.
[We've just performed a proof using the technique called mathematical induction.]
Back to the questions we had to answer:
On the 12th day, he gives away 2^12 - 1 = 4096 - 1 = 4,095 lollipops.
On the 24th day, he gives away 2^24 - 1 = 16,777,215 lollipops.
These are just particular cases of the general rule:
On the nth day, he gives away 2^n - 1 lollipops.