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.
$$\frac{\sin\frac{a + c}{3}\sin\frac{nb}{2}}{\sin{nb/2}}$$