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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(a)}
Assumptions:nZ0  and  aC  and  bC  and  aπZn \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}
\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}
Fungrim symbol Notation Short description
Sumnf(n)\sum_{n} f(n) Sum
Sinsin(z)\sin(z) Sine
ZZGreaterEqualZn\mathbb{Z}_{\ge n} Integers greater than or equal to n
CCC\mathbb{C} Complex numbers
Piπ\pi The constant pi (3.14...)
ZZZ\mathbb{Z} Integers
Source code for this entry:
    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))))

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2021-03-15 19:12:00.328586 UTC