Let a is the first term and d is the common difference of the AP

Now, the series is:

a, a + d, a + 2d, a + 3d, a + 4d, ................up to 2n terms

Now, even position series is:

a + d, a + 3d, a + 5d, ............up to n terms

Now, Sum = (n/1) * {2(a + d) + (n - 1)2d}

=> p = (n/2) * {2(a + d) + (n - 1)2d}  .................1

Again, odd position series is:

a, a + 2d, a + 4d, ............up to n terms

Now, Sum = (n/2) * {2a + (n - 1)2d}

=> q = (n/2) * {2a + (n - 1)2d} ..............2

Divide equation 1 and 2, we get

      p/q = [(n/2) * {2(a + d) + (n - 1)2d}]/[(n/2) * {2a + (n - 1)2d}]

=> p/q = {2(a + d) + (n - 1)2d}/{2a + (n - 1)2d}

=> p{2a + (n - 1)2d} = q{2(a + d) + (n - 1)2d}

=> 2ap + 2pd (n - 1) = 2qa + 2qd + (n - 1)2qd

=> 2ap +2pd (n - 1) = 2qa + d{2q + q(n - 1)}

=> 2ap - 2qa = d{2q + q(n - 1)} - 2pd (n - 1)

=> 2a(p - q) = d{2q + qn - q - 2pn + 2p}

=> 2a(p - q) = d{q + qn - 2pn + 2p}

=> d = {2a(p - q)}/{q + qn - 2pn + 2p}

Now, n^th term of the series is:

      T_n = a + (n - 1)d

=> T_n = a + (n - 1) * [{2a(p - q)}/{q + qn - 2pn + 2p}]

=> T_n = a[1  + (n - 1) * {2(p - q)}/{q + qn - 2pn + 2p}]

=> T_n = a[{q + qn - 2pn + 2p  + (n - 1) * 2(p - q)}]/{q + qn - 2pn + 2p}