Math
posted by Matt
Penn writes a 2013term arithmetic sequence of positive integers, and Teller writes a different 2013term 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?

Steve
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(PdTd) = 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. 
Billybob
I got 503.....
Respond to this Question
Similar Questions

Math
A sequence is formed by adding together the corresponding terms of a geometric sequence and and an arithmetic sequence.The common ratio of the geometric sequence is 2 and the common difference of the arithmetic sequence is 2.The first … 
Math *URGENT
Please give the answers and solutions for each. 1.If the second term is 2 and the seventh term of a geometric sequence is 64, find the 12th term. 2. Which term if the geometric sequence 18,54,162,486,... is 3,188,646? 
Maths
1..The first 2 terms of a geometric progression are the same as the first two terms of an arithmetic progression.The first term is 12 and is greater than the second term.The sum of the first 3 terms od the arithmetic progression is … 
Math
Alpha writes the infinite arithmetic sequence 10, 8, 6, 4, 2, 0... Beta writes the infinite geometric sequence 9, 6, 4, 8/3, 16/9,... Gamma makes a sequence whose n^th term is the product of the n^th term of Alpha's sequence and the … 
Algebra
True or False 1. – 5, – 5, – 5, – 5, – 5, … is an arithmetic sequence. 2. In an arithmetic sequence, it is possible that the 13th term is equal to its 53rd term. 3. In an arithmetic sequence, the common difference is computed … 
Math
Penn writes a 2013term arithmetic sequence of positive integers, and Teller writes a different 2013term arithmetic sequence of integers. Teller's first term is the negative of Penn's first term. Each then finds the sum of terms in … 
math
in an arithmetic sequence the common difference is equal to 2.the first term is also the first term of a geometric sequence. the sum of the first 3 terms of an arithmetic sequence and the sum of the first 9 terms of an arithmetic sequence … 
mathematics
1.1 the third term of an arithmetic sequence is 8 and the 15 term is 44 . Calculate: 1.1 the common difference and first term. 1.2 the sum of the first 50 terms. 1.2 1;4;7;10.... Is an arithmetic sequence .find: 1.2.1 the 30 term. 
Algebra
I am so lost on these problems. Write a geometric sequence that starts with 3 and has a common ratio of 5. What is the 23rd term in the sequence. Write an arithmetic sequence that has a common difference of 4 and the eighth term is … 
Quick math help
Find the sum of the first 10 positive integers Is it 110?