Penn writes a 2013-term arithmetic sequence of positive integers, and Teller writes a different 2013-term arithmetic sequence of integers. Teller's first term is the negative of Penn's first term. Each then finds the sum of terms in his sequence. If their sums are equal, then what is the smallest possible values of the first term in Penn's sequence?
I got 503.....
Let's assume the first term of Penn's sequence is 'a'. Since Teller's first term is the negative of Penn's first term, Teller's first term is '-a'.
The sum of an arithmetic sequence can be found using the formula:
S = (n/2) * (2a + (n - 1)d)
where S is the sum, a is the first term, n is the number of terms, and d is the common difference.
For Penn's sequence, the sum is equal to:
S1 = (2013/2) * (2a + (2013 - 1)d)
For Teller's sequence, the sum is equal to:
S2 = (2013/2) * (-2a + (2013 - 1)d)
Since the sums are equal, we can set them equal to each other and solve for 'a':
(2013/2) * (2a + (2013 - 1)d) = (2013/2) * (-2a + (2013 - 1)d)
Dividing both sides by (2013/2) gives us:
2a + (2013 - 1)d = -2a + (2013 - 1)d
Rearranging this equation:
4a = 0
Dividing by 4 gives us:
a = 0
Therefore, the smallest possible value of the first term in Penn's sequence is 0.
To find the smallest possible value of the first term in Penn's sequence, we need to understand the problem and use some algebraic reasoning.
Let's call Penn's first term "p" and Teller's first term "-p" (since it is the negative of Penn's first term).
We know that both Penn and Teller have an arithmetic sequence of 2013 terms. An arithmetic sequence has a common difference between each term, so let's call that common difference "d".
The sum of an arithmetic sequence can be found using the formula: sum = (n/2)(first term + last term), where "n" is the number of terms in the sequence.
For Penn, the sum of the terms in his sequence would be: (2013/2)(p + p + (2012)d) = 2013p + 2012d.
For Teller, the sum of the terms in his sequence (using the negative sign for the first term) would be: (2013/2)(-p + (-p) + (2012)d) = -2013p + 2012d.
According to the problem, Penn and Teller have equal sums:
2013p + 2012d = -2013p + 2012d.
By simplifying the equation, we can cancel out the common terms:
2(2013p) = 0
This equation shows that 2 multiplied by 2013p equals zero.
To find the smallest possible value of "p", we want to make the coefficient of p as small as possible. In this case, the coefficient is 2013, but since p is positive, the smallest possible value would be when p = 0.
Therefore, the smallest possible value of the first term in Penn's sequence is 0.
If the sequences are P and T, and their differences are Pd and Td, then
T1 = -P1
2013/2 (2T1+2012Td) = 2013/2 (2P1 + 2012Pd)
or,
2013P1 + 2025078Pd = 2013T1 + 2025078Td
4026P1 + 2025078(Pd-Td) = 0
We know that since T1 < 0 and the sum of T is the same as the sum of P, that Td > Pd. What if Td = Pd+1? Then we have
4026P1 = 2025078
P1 = 503
What if Td = Pd+k, with k>0?
4026P1 = 2025078k
P1 = 503k
Let's check.
P = 503,504,...,2515
Sum = 2013/2 (503+2515) = 3037617
T = -503,-501,...,3521
Sum = 2013/2 (-503+3521) = 3037617
cool problem.