Arithmetic Progression - two series I have two arithmetic progressions: $a, b, c, d$ and $w, x, y, z$
If the arithmetic progressions are merged together like this:
$aw, bx, cy, dz$, is it possible to find the sum of the series?
Let $a$ be the first term and $c$ be the last term of the series.
Let $n$ be the number of terms in the series and $b$ the common difference.
$$\frac{\sin\frac{a + c}{3}\sin\frac{nb}{2}}{\sin{nb/2}}$$


I have two arithmetic progressions: $a, b, c, d$ and $w, x, y, z$
  If the arithmetic progressions are merged together like this:
  $aw, bx, cy, dz$, is it possible to find the sum of the series?

The result is $$(b+c)(x+y)+5(c-b)(y-x),
$$
or, equivalently, $$2b(3x-2y)+2c(3y-2x),
$$
or, equivalently, $$2x(3b-2c)+2y(3c-2b),
$$
or, finally,
$$
6(bx+cy)-4(by+cx).$$
Proof: Use $a=2b-c$, $d=2c-b$, $w=2x-y$, $z=2y-x$, and simplify.
