Summation notation help
- Thread starter Falksong
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Hi, please could someone help guide me in the right direction with this equitation? No idea where to start ?
Thanks
Falksong
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What is \(\displaystyle x_{1} = \;\;\)? How about \(\displaystyle x_{2} = \;\;\)?
Harry_the_cat
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Do you know what the symbols \(\displaystyle \Sigma\) and \(\displaystyle \Pi\) mean?
Hi, so I get Sigma and I guess I treat Pi as the sum of Pi x in the equation? I'm also confused about the sum of x in the brackets, :-/ please might you be able to break it down for me? ?
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pka
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Hi, please could someone help guide me in the right direction with this equitation? No idea where to start ?
Thanks
Falksong
\(\displaystyle \prod\limits_{k = 1}^6 {\left( {{x_k}} \right)} = \left( {{x_1}} \right)\left( {{x_2}} \right)\left( {{x_3}} \right)\left( {{x_4}} \right)\left( {{x_5}} \right)\left( {{x_6}} \right) = ?\)
Use your calculator to finish. POST YOUR ANSWER.
D
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Harry_the_cat
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The symbol \(\displaystyle \Sigma\) (upper case Greek letter sigma) stands for "sum of". Sigma = Sum. To find the sum, you add.
The symbol \(\displaystyle \Pi\) (upper case Greek letter pi) stands for "product of". Pi = Product. To find the product, you multiply.
\(\displaystyle x_1\) is the first number in the set X. Here \(\displaystyle x_1 = 5.2\).
\(\displaystyle x_2\) is the second number in the set X, etc.
So here, you simply need to multiply all the numbers from \(\displaystyle x_1\) to \(\displaystyle x_6\). That's what the \(\displaystyle i=1\) and the \(\displaystyle 6\) are telling you.
In your other post, you need to add all the terms.
The maths here (arithmetic) is easy, but you've got to be able to read the notation.
Steven G
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Are you surprised at that?
pka
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I can understand why one might not know a particular bit of notation.
However, once the notation is explained along with detailed examples if the person says "Still way over my head"; then I say to him/her you will drown if you do not go back and do the necessary prerequisites.
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Otis
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No -- you don't use pi or add numbers, in this exercise.
The Greek alphabet contains upper-case letters and lower-case letters, just like the English alphabet. Note the differences.
The Greek upper-case letter Pi looks like this: \(\displaystyle \Pi\)
The lower-case letter pi looks like this: \(\displaystyle \pi\)
The upper-case letter Sigma looks like this: \(\displaystyle \Sigma\)
The lower-case letter sigma looks like this: \(\displaystyle \sigma\)
In mathematics, each of these symbols has a different meaning.
Sigma notation \(\displaystyle \Sigma\) is used for summations; we add terms (elements in a set).
Pi notation \(\displaystyle \Pi\) is used for products; we multiply terms.
pi \(\displaystyle \pi\) represents the constant 3.14159265… (i.e., the ratio of a circle's circumference to its diameter).
sigma \(\displaystyle \sigma\) represents standard deviation (in statistics).
Of course, each of these symbols can be defined to represent other meanings in different contexts, but the definitions above are what you'll see most often (i.e., standard meanings).
The Pi notation in your exercise instructs you to multiply together terms 1 through 6, from the given set called x.
There is no summation, in this exercise, because there is no \(\displaystyle \Sigma\).
You just said you "get" Sigma notation. It seems like you're not quite there, yet.
For more exposure to the various patterns (eg: worked examples and practice exercises), google keywords Sigma notation and also Pi notation. Watch some videos; read some lessons; write out some worked examples (as you follow along); do extra practice problems.
If you see anything you don't understand, while studying, come back and ask us about it.
Cheers :cool:
Otis
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That's not a helpful explanation. Instead, please explain specifically what you're confused about.
Is it the subscripted symbols?
Is it the switch to using symbol k for the index, instead of the given symbol i ?
Maybe it's the concept of an index?
Something else (like all of the above, heh)?
If you find yourself unable to articulate your confusion, then you need to back up and review. Google some keywords, and watch some videos. Learn and practice the patterns. :cool:
PS: Please check out the forum's submission guidelines. Cheers.
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Hi All,
Hi Everyone, thank you for all your replies and apologies if I was unclear.
My confusion is, What to do with X = [5.2,3.4,8.9,9.00,2.2,4.6] .. from your threads, am I correct in understanding that I multiply these all together to get x = 14331.62? Then Multiply X x 1, X x2.... X x 6?
If thats right, then is my calculation in the snip below correct? Given
Hi Everyone, thank you for all your replies and apologies if I was unclear.
My confusion is, What to do with X = [5.2,3.4,8.9,9.00,2.2,4.6] .. from your threads, am I correct in understanding that I multiply these all together to get x = 14331.62? Then Multiply X x 1, X x2.... X x 6?
If thats right, then is my calculation in the snip below correct? Given
| 6 | |
| Π | 14331.62 |
| 1 |
HallsofIvy
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I get 14331.62016.
NO! Those are just "indices" labeling each number. One could complain that we are not actually told which number is \(\displaystyle x_1\), which is \(\displaystyle x_2\), etc. but since they are all multiplied together, it doesn't matter.
Well you truncated the last 3 decimals. The exact answer is 14331.62016 as you would have found had you used a hand calculator or changed the format of your cells to show five decimals rather than just two. But you were not required to get an exact answer so your answer is good enough.
The next part is totally unnecessary.
Let's go over this notation
\(\displaystyle \displaystyle \prod_{i=1}^6x_i.\)
represents a single number, in this case 14331.62016, that is the result of multiplying a number of expressions. As Harry explained, the P in Pi stands for Product.
You must think of i as an ordered set of numbers starting with 1 (because it says i = 1 at the bottom of the capital Pi), increasing each time by 1, and ending with 6 (because it says 6 at the top of the capital Pi). In other words, i starts with 1, follows with 2, then 3, then 4, then 5, and ends with 6. So
\(\displaystyle \displaystyle \prod_{i=1}^6 x_i \equiv x_1 * x_2 * x_3 * x_4 * x_5 * x_6.\)
In that example, i is used only for an index. But that is not necessarily true. For example,
\(\displaystyle \displaystyle \mathbb X \ = \{2,\ 4,\ 6,\ 8\} \text { and } y = \left ( \prod_{i=1}^4 (5 - i)x_i \right ) \implies\)
\(\displaystyle y \equiv \{(5 - 1) * 2\} * \{(5 - 2) * 4\} * \{(5 - 3) * 6\} * \{(5 - 4) * 8\ =\)
\(\displaystyle (4 * 2)(3 * 4)(2 * 6)(1 * 8) = 8 * 12 * 12 * 8 = 2^{10} * 3^2 = 1024 * 9 = 9216.\)
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