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.
$$\frac{\sin\frac{a + c}{3}\sin\frac{nb}{2}}{\sin{nb/2}}$$
Best Answer
The $i^{\text{th}}$ of the first series (if we start counting at zero) is $a+i(b-a)$. The $i^{\text{th}}$ term of your combined series is then $[a+i(b-a)][w+i(x-w)]=aw+i[a(x-w)+w(b-a)]+i^2(b-a)(x-w)$ Now you can use the sum of powers formulas to sum over the range of $i$ you desire.