2014 numbers are placed along the circumference of a circle. When any five successive numbers are added, the total is always 40. What are these 2014 numbers?
Please explain how you figured this out. Working is really appreciated.
Looks to me like they could all be 8. Are there any restrictions on the values allowed?
2014 is not a multiple of 5, so there is no cycle of 5 numbers which repeats.
To solve this problem, we need to find a pattern that allows us to determine the 2014 numbers. Let's break down the problem step by step:
1. Let's assume the first number in the sequence is X.
2. We know that any five successive numbers always sum up to 40. So, if we take any five consecutive numbers in the circle, the sum would be:
X + (X+1) + (X+2) + (X+3) + (X+4) = 40
3. We can simplify this equation:
5X + 10 = 40
5X = 30
X = 6
So, the first number (X) is 6.
4. Now, we need to find the next four numbers. To do this, we add 1 to the previous number (since the numbers are consecutive).
The sequence would be:
6, 7, 8, 9, 10
5. We can continue this pattern, finding the next four numbers each time:
11, 12, 13, 14, 15
16, 17, 18, 19, 20
and so on...
6. However, we need to consider that this sequence will eventually wrap around the circle after reaching the number 40 (since the sum of any five numbers is always 40). Therefore, the numbers will repeat.
So, the sequence will be:
6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40.
We can see that each subsequent set of five numbers will be the same as the first five (6, 7, 8, 9, 10). So, we can repeat this sequence until we reach the 2014th number.
Therefore, the 2014 numbers will be:
6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 6, 7, 8, 9, 10, ...
Keep in mind that these are the numbers in a circle, so they are not in a linear order.