Tile Pattern...
A design starts with 1 tile and it continues to grow the next day to 4 tiles and on the third day there is 13 tiles. How many tiles will there be on the 10th day? How many tiles on the 50th day? Find a formula for any number?
52.2600.
Since 1 , 4, 13 do not form either an arithmetic nor a geometric sequence,
having only 3 terms is not enough information to see the pattern
e.g. I could claim that the sequence is
1 4 13 40 121 by claiming that I add increasing powers of 3 to get the next term
that is,
1+3 = 4
4+9 = 13
13+27 = 40
40 +81 = 121
To find the number of tiles on each day, let's observe the pattern:
Day 1: 1 tile
Day 2: 4 tiles
Day 3: 13 tiles
If we look closely, we can see that the number of tiles on each day increases as follows: 1, 4, 13.
To find the number of tiles on the next day, we need to look for a pattern in the differences between the numbers:
4 - 1 = 3
13 - 4 = 9
The differences between consecutive numbers are 3 and 9. It seems like the difference between consecutive numbers is increasing by 6 each time.
So, let's continue this pattern to find the number of tiles on the 4th day:
13 + 9 = 22 tiles
And on the 5th day:
22 + 15 = 37 tiles
As we add 6 to the difference each time, we can derive a formula to calculate the number of tiles on any given day:
Number of tiles on day n = n^2 + (n-1) * 3
Now, let's use this formula to find the number of tiles on the 10th and 50th days:
Number of tiles on the 10th day = (10^2) + (10-1) * 3
= 100 + 9 * 3
= 100 + 27
= 127 tiles
Number of tiles on the 50th day = (50^2) + (50-1) * 3
= 2500 + 49 * 3
= 2500 + 147
= 2647 tiles
Therefore, on the 10th day, there will be 127 tiles, and on the 50th day, there will be 2647 tiles.