Penn writes a 2013-term arithmetic sequence of positive integers, and Teller writes a different 2013-term arithmetic sequence of integers. T

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 the terms in his sequence. If their sums are equal, then what is the smallest possible value of the first term in Penn's sequence?

Arithmetic series Penn: $$\small{ \text{ $p_n = p_1 + (n-1) d_p $ }}\ . \quad \text{The sum} \ \text{ $ s_p= \frac{n}{2}*[2p_1+(n-1)d_p] $ }}$$  

Arithmetic series Teller: $$\small{\text{$t_n = t_1 + (n-1) d_t $ }}\ . \quad \text{The sum} \ \text{ $ s_t= \frac{n}{2}*[2t_1+(n-1)d_t] $ }}$$

I.

$$\small{\text{ $ s_p=s_t $ }} $\\$ \small{\text{ $\frac{n}{2}*[2p_1+(n-1)d_p]=\frac{n}{2}*[2t_1+(n-1)d_t] $ }} $\\$ \small{\text{ $2p_1+(n-1)d_p = 2t_1+(n-1)d_t $ }}$$

II.

$$\small{\text{ $ t_1= -p_1 $ }} $\\$ \small{\text{ $2p_1+(n-1)d_p = -2p_1 +(n-1)d_t $ }} $\\$ \small{\text{ $4p_1 = (n-1)d_t - (n-1)d_p = (n-1)(d_t-d_p) $ }}$$

III.

$$\small{\text{ $ n=2013$ }} $\\$ \small{\text{ $4p_1 = 2012(d_t-d_p)$ }} $\\$ \small{\text{ $p_1 = 503(d_t-d_p)$ }}$$

IV.

The smallest possible value of the first term in  Penn's sequence $$\small{\text{$p_1$}}$$  is 503, if $$\small{\text{$(d_t-d_p) = 1 $}}$$

I'm not sure I get it. Both Penn and Teller have sequences with positive integers, yet Teller's first term is the negative of Penn's first term. That sounds contradicting to me ...?

Very nice, heureka...........

Thanks very much Heureka  

I missed the bit about there being 2013 terms    doh!