the design below grows each day, as shown. if the pattern continues to grow like this, how many tiles will there be in the design on the tenth day? on the fiftieth day?
Day 1 1 tile
Day 2 5 tiles
Day 3 13 tiles
Day 3 does not fit: 1+4*2 = 9
Since the difference is changing by 4 each day, we will have a quadratic:
Day n: 2n^2 - 2n + 1
1+4(n-1) where n is the day number
on day 10
1+4(10-1)=37
on day fifty
1+4(50-1)=197
What do you mean by the ^ in your equation Steve? I don't really understand the 2n^2-2n+1. I did figure out it was a quadratic equation, and actually happened to figure out the entire problem until I noticed it said "Write a description or formula that allows me to figure out the number of tiles for any day number?"
Oh, ok, do you mean to cube the first number by the 2nd, get back to me... Thnx
To find the number of tiles in the design on the tenth and fiftieth day, we can observe the pattern and try to come up with a formula to calculate the number of tiles on each day.
If we carefully examine the number of tiles each day, we can see that it follows a specific pattern. On each day, the number of tiles in the design is increasing by an odd number.
Let's break down the number of tiles on each day:
Day 1: 1 tile
Day 2: 1 + 4 = 5 tiles (increased by 4)
Day 3: 5 + 8 = 13 tiles (increased by 8)
We can notice that the increase in the number of tiles on each day is equal to the corresponding term of an arithmetic sequence. The common difference between the terms of the sequence is as follows:
1st term: 4 (the increase on day 2)
2nd term: 8 (the increase on day 3)
3rd term: 12 (the increase on day 4)
...
We can now calculate the common difference by finding the difference between the increase on day 2 and day 1, which is 4 - 0 = 4. Therefore, the common difference is 4.
Now, let's determine the formula to calculate the number of tiles on any given day. Since the first term (day 1) is 1 tile, we can express the number of tiles on the nth day using the arithmetic sequence formula:
nth term = a + (n - 1) * d
Where:
nth term is the number of tiles on the nth day,
a is the first term (1 tile),
n is the day number, and
d is the common difference (4).
Applying this formula, we can find the number of tiles on the tenth and fiftieth day.
On the tenth day:
nth term = 1 + (10 - 1) * 4
= 1 + (9) * 4
= 1 + 36
= 37 tiles
On the fiftieth day:
nth term = 1 + (50 - 1) * 4
= 1 + 49 * 4
= 1 + 196
= 197 tiles
Therefore, on the tenth day, there will be 37 tiles in the design, and on the fiftieth day, there will be 197 tiles.