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The formula for an arithmetic series is $S_n=\dfrac{n}{2}[2a+(n-1)d]$. With $n\in\mathbb{Z}^{+}$. $a$ is the first term of the sequence and $d$ is the common ratio.

My question is does this formula work if both $a, d\in\mathbb{R}$, or do they have to be integers?

hardmath
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Andres Romero
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1 Answers1

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It works with real numbers, complex numbers, rational numbers, quaternions, etc. It involves only finite sums, so it is valid for rings such as polynomials, matrices, etc. and even for structures like vector spaces that only have vector addition and scalar multiplication.

As written the one limitation involves division by two, which is valid except where the "scalar field" is characteristic two. If you need more information about that exception, just ask.

hardmath
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  • what is a scalar field – Saketh Malyala Jun 19 '17 at 01:59
  • A field is an algebraic setting where we can divide by nonzero values. The word scalar identifies the values that a vector space allows to "scale" the sizes of vectors. Basically dividing by two is multiplying by $1/2$, but there are fields in which $2=0$ and we cannot divide by two in this way. However the formula given above could be rewritten to a form valid even in that case. – hardmath Jun 19 '17 at 02:03