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"),
    Formula(Equal(Sum(Sin(Add(Mul(Mul(2, a), k), b)), For(k, 0, n)), Div(Mul(Sin(Mul(a, Add(n, 1))), Sin(Add(Mul(a, n), b))), Sin(a)))),
    Variables(a, b, n),
    Assumptions(And(Element(n, ZZGreaterEqual(0)), Element(a, CC), Element(b, CC), NotElement(Div(a, Pi), ZZ))))

Topics using this entry

Sine