The first three elements of a number sequence are 1, 2, 3. ( As you can see, the middle number is one less than the product of its neighbors. This is true for the rest of the elements of this sequence. Now add the elements of the sequence up to a certain element. Could this sum be 2009? Could this sum be 2010?
the sequence runs this way :
1 2 3 2 1 1 2 3 ..
it repeats!
the first loop is 1 2 3 2 1
the sum of that loop is 9
so as long as we add up to the end of a loop we have multiples of 9
If we add up 223 loops the sum is 2007
adding the next digit, which is a 1 would give up 2008, and if we add the next one after that, we would get 2010
So it is not possible for a sum of 2009, but we will reach 2010
THANK YOU! =)
To determine whether the sum of the elements of the sequence could be 2009 or 2010, we need to find a pattern in the sequence and calculate the sum up to a certain element.
The given sequence begins with 1, 2, 3, and the pattern states that each middle number is one less than the product of its neighbors. Let's continue the sequence to better understand the pattern:
1, 2, 3, 2, 6, 5, 30, 29, 870, 869, ...
We can observe that each element after the initial three follows the pattern: the middle number is one less than the product of its neighbors.
Now let's calculate the sum of the sequence up to a certain element. To accomplish this, we need to add all the elements in the sequence up to the desired position.
For example, if we want to know the sum up to the 5th element, we would calculate: 1 + 2 + 3 + 2 + 6 = 14.
Now, let's calculate if the sum could be 2009:
1 + 2 + 3 + 2 + 6 + 5 + 30 + 29 + 870 + 869 + ... = sum
Since the sequence continues indefinitely, we cannot directly calculate the sum. However, we can certainly calculate the sum up to a certain point. As we continue adding elements to the sequence, the sum will increase proportionally.
To check if the sum could be 2009, we need to calculate the sum up to a point where the sum is greater than 2009 but still close to that value. Let's calculate the sum up to the 10th element:
1 + 2 + 3 + 2 + 6 + 5 + 30 + 29 + 870 + 869 = 1817
We see that the sum up to the 10th element is less than 2009. This means that the sum of the sequence cannot be equal to 2009.
Now, let's check if the sum could be 2010:
Using the same method, let's calculate the sum up to the 10th element:
1 + 2 + 3 + 2 + 6 + 5 + 30 + 29 + 870 + 869 = 1817
We observe that the sum up to the 10th element is still less than 2010.
To determine whether the sum can reach 2010, we need to continue adding elements until the sum exceeds 2010. Let's calculate the sum up to the 11th element:
1 + 2 + 3 + 2 + 6 + 5 + 30 + 29 + 870 + 869 + ... = sum
Now we see that the sum has exceeded 2010, so it is possible for the sum to be equal to 2010.
Therefore, the sum of the elements in this sequence can be 2010, but it cannot be 2009.