# Fungrim entry: b8ab9c

$\sum_{k=0}^{n} \sin\!\left(2 a k + b\right) = \frac{\sin\!\left(a \left(n + 1\right)\right) \sin\!\left(a n + b\right)}{\sin(a)}$
Assumptions:$n \in \mathbb{Z}_{\ge 0} \;\mathbin{\operatorname{and}}\; a \in \mathbb{C} \;\mathbin{\operatorname{and}}\; b \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \frac{a}{\pi} \notin \mathbb{Z}$
TeX:
\sum_{k=0}^{n} \sin\!\left(2 a k + b\right) = \frac{\sin\!\left(a \left(n + 1\right)\right) \sin\!\left(a n + b\right)}{\sin(a)}

n \in \mathbb{Z}_{\ge 0} \;\mathbin{\operatorname{and}}\; a \in \mathbb{C} \;\mathbin{\operatorname{and}}\; b \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \frac{a}{\pi} \notin \mathbb{Z}
Definitions:
Fungrim symbol Notation Short description
Sum$\sum_{n} f(n)$ Sum
Sin$\sin(z)$ Sine
ZZGreaterEqual$\mathbb{Z}_{\ge n}$ Integers greater than or equal to n
CC$\mathbb{C}$ Complex numbers
Pi$\pi$ The constant pi (3.14...)
ZZ$\mathbb{Z}$ Integers
Source code for this entry:
Entry(ID("b8ab9c"),
Assumptions(And(Element(n, ZZGreaterEqual(0)), Element(a, CC), Element(b, CC), NotElement(Div(a, Pi), ZZ))))