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I need to choose weather this is a product notation or a summation. I can figure out which one it is.

I have this expression:

$$2 \times 4 \times 6 \times 8 \times 10 \ldots \times 40$$

The answer is either:

$$\sum_{m=2}^{40} m$$

or

$$\prod_{m=2}^{40} m$$

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    $\begingroup$ Mathjaxed for future reference. Neither is correct though. $\endgroup$ Oct 25, 2016 at 1:35
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    $\begingroup$ FYI, "product" is just a fancy name for "result of multiplying stuff together". $\endgroup$
    – jpmc26
    Oct 25, 2016 at 2:45
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    $\begingroup$ I'm genuinely curious. They taught you about the product notation, but they didn't teach you that "product" means "multiplication"? Or they didn't teach you that $\times$ is multiplication? $\endgroup$ Oct 25, 2016 at 10:39

4 Answers 4

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Neither of your proposal is correct.

For your first guess, it means $$2+3+4+\ldots+ 40$$

For your second guess, it means $$2(3)(4) \ldots (40)$$

You are multiplying even numbers, it should be

$$\prod_{i=1}^{20} (2i)$$

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You are multiplying values, so you should probably use the product notation:

$$\prod_{m=1}^{20}2m = 2 \times 4 \times 6 ... \times 40$$

When you have something like $\prod_{m=2}^{40}m$ as in your example, this actually represents the product $2 \times 3 \times 4 ... \times 39 \times 40$ - it includes the odd numbers too, since $m$ increases by $1$ each time. To increase it by $2$, use $(2m)$ in the product instead of just $m$.

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Can also be expressed as $$2^{20} 20!$$

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  • $\begingroup$ :D most helpful $\endgroup$
    – image357
    Oct 25, 2016 at 8:51
  • $\begingroup$ @Marcel - Thank you :) $\endgroup$ Oct 25, 2016 at 8:52
  • $\begingroup$ An alternative notation is sometimes used: the double factorial $40!!$, where $n!!$ means the product of all positive integers less than or equal to $n$ and of the same parity as $n$, that is, $2\times4\times6\times\cdots\times n$ if $n$ is even and $1\times3\times5\times\cdots\times n$ if $n$ is odd. $\endgroup$
    – David K
    Oct 25, 2016 at 13:32
  • $\begingroup$ Yes that's right. I wonder if a triple factorial exists, e.g. $9!!!=9\times 6\times 3$... $\endgroup$ Oct 25, 2016 at 13:34
  • $\begingroup$ @hypergeometric : Yes. See also multifactorials, where a slightly less awful notation is shown. $\endgroup$ Oct 25, 2016 at 14:55
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Neither is correct. It is in fact the product notation, as summation notation would be $2+4+6+8+...+40$, however, the expression is incorrect. The correct expression is $\prod_{m=1}^{20}2m$. There is a way to do it with summation notation, but I don't think that's what you're looking for.

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