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Fungrim entry: b8ab9c

k=0nsin ⁣(2ak+b)=sin ⁣(a(n+1))sin ⁣(an+b)sin ⁣(a)\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\!\left(a\right)}
Assumptions:nZ0andaCandaπZn \in \mathbb{Z}_{\ge 0} \,\mathbin{\operatorname{and}}\, a \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\!\left(a\right)}

n \in \mathbb{Z}_{\ge 0} \,\mathbin{\operatorname{and}}\, a \in \mathbb{C} \,\mathbin{\operatorname{and}}\, \frac{a}{\pi} \notin \mathbb{Z}
Definitions:
Fungrim symbol Notation Short description
Sumnf ⁣(n)\sum_{n} f\!\left(n\right) Sum
Sinsin ⁣(z)\sin\!\left(z\right) Sine
ZZGreaterEqualZn\mathbb{Z}_{\ge n} Integers greater than or equal to n
CCC\mathbb{C} Complex numbers
ConstPiπ\pi The constant pi (3.14...)
ZZZ\mathbb{Z} Integers
Source code for this entry:
Entry(ID("b8ab9c"),
    Formula(Equal(Sum(Sin(Add(Mul(Mul(2, a), k), b)), Tuple(k, 0, n)), Div(Mul(Sin(Mul(a, Add(n, 1))), Sin(Add(Mul(a, n), b))), Sin(a)))),
    Variables(a, n),
    Assumptions(And(Element(n, ZZGreaterEqual(0)), Element(a, CC), NotElement(Div(a, ConstPi), ZZ))))

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Copyright (C) Fredrik Johansson and contributors. Fungrim is provided under the MIT license. The source code is on GitHub.

2019-08-21 11:44:15.926409 UTC