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Use sigma notation to write the sum 5 + 9)Ke) +/3(5 + 52)%)...

Question

Use sigma notation to write the sum 5 + 9)Ke) +/3(5 + 52)%)

Use sigma notation to write the sum 5 + 9)Ke) +/3(5 + 52)%)



Answers

Use sigma notation to write the sum. $$\frac{5}{1+1}+\frac{5}{1+2}+\frac{5}{1+3}+\dots+\frac{5}{1+15}$$

Now let's look at a now let's look at a question where they give us an expanded expression and they want us to write it back as a summation notation. They don't actually want us to evaluate it. They want us to write it as the summation notation that it came from. Now when you're doing this, you want to look from term to term. So whatever separated by your pluses in here are going to be your separate terms. So I've won over five times one plus 1/5 times two. So my first term is 1/5 times one. Then my second term is 1/5 times two plus all the way up to and then 1/5 times 11. So here I want to look and see what is consistently the same, that 1/5. And then what is implementing? So here is what the five is multiplied to in the denominator. So that's the part that is actually the replacements for your index of summation. So if we pick whatever index of some nation we want, let's say we want to use the index of some nations BK So some nation K is equal to. Well the plug ins, the things that are changing as they go. It starts at one And then to gets plugged in and then all the way up to 11. So that's how we decide about our values there. And then what expression is it being plugged into? Well, they all have a one in the numerator, a fraction. Bar they all have a factor of five in the denominator. And what the five is multiplied to is the thing that's implementing us. So that's where your index of summation K goes. So the summation that we can write to represent that expanded form is the summation cable 1 to 11 of 1/5 K.

In discussion. We need to write be given some as the sigma notation. So we can see the terms of this uh some the difference of these terms And the succeeding term is three. So it can be done as this one can be done is one plus three in 2, zero four grand millionaires. One plus three multiplied by one 7. 10 million has one plus three multiplied by two and so on. The last time is 37. This can be done as one plus three, multiplied by 12 Which is 20 to 36 plus one is 37. Now we can see this this one is Similar in all the terms and also these three similar But this year zero is increasing uh in each succeeding term by one. So this zero then one then two and the last one is tell so this is the variable tongue which can be assigned as key. So the general term, general term for this some can be done is one plus three key. We're case wearing from 0 to 12. Now this some can be redone as the sigma notation as the sigma which means summation of one plus tiki. We are case from 0 to well. So this is the sigma notation for this song. I hope all of you go discussion. Thank you. Yeah

Now let's look an example where they give us an expanded expression and they would like us to write it as the summation notation. That would expand out to be that expression. So here I have in the bracket seven times 1/6 plus five. Then close the bracket plus seven times to six plus five. Close the bracket etcetera. And notice from term to term from what's in the brackets from one to the other. Um what is the same in H things? Because that will be a constant and not be incremental up with your index of summation. So in each of these I have a seven multiplied to a fraction where the denominator is six but the top is implementing so I have uh seven that's consistent in every term then times it's consistent. The denominator of six is the same in each one and then plus the five is the same in each one. The only thing that's changing is the numerator of the fraction. It's going first with a one, then a two, then a three all the way up to them. A six. So that part that's fixed will be those numbers. And that part that's incremental up will actually have our index of summation and then show what they're supposed to start with and end with with that. So we have our summation and that's representing the pluses between these groupings and I can pick whichever index of summation I want for the very or the letter part I could do I or K or J or whatever I'm going to use. I This isn't the complex number I this is just the index of summation. I and a lot of times they do use I because it is the first letter of the word index equals look at the enumerators first it's a one, then a two, then a three all the way up to a six, so I equals one for the start and six for the end. And then the expression is bracket seven times. The numerator is what's implementing. So that's my eye over the denominators six each time. So I just leave that to six and then close that parentheses plus five and close the bracket. And there is your summation notation for that given expanded expression.

Okay so here we're giving the somewhere we have a someone and then we are adding um a sub three and then we are adding a sub five and then we are adding a sub seven. And then this pattern keeps continuing And then up to we are adding a sub 99. So um here we can see that the index um is every other number requiring a factor like like um like to I but we also see that our indexes are all on right so therefore we need to to the I -1 to give all our index is odd. So therefore this is going to be the sum where we have I while starting at one and they're gonna be 50 terms here. So I go from one 250 of just a um sub to I Uh to I -1. Take care.


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