{"id":4746,"date":"2010-06-15T13:41:59","date_gmt":"2010-06-15T05:41:59","guid":{"rendered":"http:\/\/www.intmath.com\/blog\/?p=4746"},"modified":"2014-11-21T18:04:47","modified_gmt":"2014-11-21T10:04:47","slug":"summation-notation","status":"publish","type":"post","link":"https:\/\/www.intmath.com\/blog\/mathematics\/summation-notation-4746","title":{"rendered":"Summation notation"},"content":{"rendered":"

I had the following exchange with a reader over the last few weeks.<\/p>\n

It's long (over 2 parts), but if you want to understand summation notation, it's worth reading!<\/p>\n

This post also demonstrates the Socratic Approach to math teaching, where the teacher gives space for the learner to figure things out for themselves.<\/p>\n

The exchange was done entirely in Google Docs and I have just pasted it here.<\/p>\n

Note how respectful Yousuf is. I don't mind helping him at all as he is willing to show his appreciation at every step. Sadly, not everyone responds like this.<\/p>\n

My replies throughout are in bold<\/b>.<\/p>\n

Yousuf asks:<\/strong><\/p>\n

Hi again sir<\/p>\n

Thanks for reading this sir, I greatly appreciate your kind support.<\/p>\n

Sir, this is the very last question on that exercise on sequences. This one's a lot different and I've a feeling it's going to be a lot more difficult especially considering the way it's written.<\/p>\n

The question is ...<\/p>\n

14. Given that:<\/b><\/p>\n

\"\\sum_{r=n+3}^{2n}{r=312}<\/p>\n

find the value of n.<\/b><\/p>\n

OK the expression's turned out kind of unusually, but just to confirm:<\/p>\n

The 2n is supposed to above the sigma sign and the r=n+3 part is supposed to be below it.<\/p>\n

I don't understand anything about this particular expression and if it's got anything to do with that previous thing we did about those sigma equations, I have no idea what to do here sir!<\/p>\n

I'd be very grateful if you could please help me on this one sir<\/p>\n

Thanks again sir, I really appreciate your time \ud83d\ude42<\/p>\n

Yousuf.<\/p>\n

\n

Me:<\/b><\/p>\n

OK. First question - write out what this means:<\/b><\/p>\n

\"\\sum_{a=1}^{5}{a^2} <\/p>\n

This will help: Helpful revision<\/a> <\/strong><\/p>\n<\/blockquote>\n

Hi again sir,<\/p>\n

It just means ...<\/p>\n

\"\\sum_{a=1}^{5}{a^2}<\/p>\n

Thanks sir<\/p>\n

Yousuf\n<\/p>\n

\n

Correct.<\/b><\/p>\n

OK, next, what is <\/b>
\n \"\\sum_{a=x+1}^{5}{(a)}<\/p>\n<\/blockquote>\n

Hi again sir,<\/p>\n

Would it be ...<\/p>\n

\"\\sum_{a=x+1}^{5}{(a)} ???????????????<\/p>\n

Thanks \ud83d\ude42<\/p>\n

Yousuf<\/p>\n

\n

Where did your 2 come from?? And how do you know it is 5 terms?<\/b><\/p>\n

Go back to the first example, which you did properly. It said \"start at a=1\" at the bottom of the Sum sign, which you did. You substituted in 1 and got 1^2.<\/b><\/p>\n

Then you added 1 and got 2, and substituted that in and got 2^2<\/b><\/p>\n

You stopped when you got to 5, since that is on the top of the Sum sign.<\/b><\/p>\n

Now, go back to my second question and start where it says to start, and finish where it says to finish.<\/b><\/p>\n<\/blockquote>\n

Hi again sir<\/p>\n

Mmmm....I first thought that we always start with 1, but it seems I'm wrong clearly.<\/p>\n

If we start at a = x+1, would it then be ...<\/p>\n

\"\\sum_{a=x+1}^{5}{(a)} <\/p>\n

And terminate at 5(x+1) since it's the last one?<\/p>\n

Thanks again sir \ud83d\ude42<\/p>\n

Yousuf<\/p>\n

\n

We are getting somewhere, but it's not quite right yet.<\/b><\/p>\n

1. There is no MULTIPLY in the question, so why are you multiplying?<\/b><\/p>\n

2. The number at the top is 5, so that should be the last number.<\/b><\/p>\n

3. Once again, you have 5 terms. How do you know it is 5 terms?<\/b><\/p>\n

Pls read my summary above about the things you did right before.<\/b><\/p>\n

Try again ^_^<\/b><\/p>\n<\/blockquote>\n

Sorry sir, but I have no idea. I'm always thinking we start off with a = 1 in all these sequences, but again I have no idea what to do here.<\/p>\n

Right, I think the first term would definately be ...<\/p>\n

\"\\sum_{a=x+1}^{5}{(a)}<\/p>\n

Because we substitute a = x+1 into (a). But how do we get to 5?<\/p>\n

What I did in the last example was that I subbed a = 1 into (a^2) and followed throughout using the natural numbers 1, 2, 3, 4 and 5 (with 5 being the last one).<\/p>\n

This one's confusing because it's x+1 which I'm not sure if I am to use the natural numbers, then this means it'll be 2, 4, 6, 8, 10???<\/p>\n

Yousuf.<\/p>\n

\n

This bit you said is quite correct:<\/b> <\/p>\n

What I did in the last example was that I subbed a = 1 into (a^2) and followed throughout using the natural numbers 1, 2, 3, 4 and 5 (with 5 being the last one).<\/p>\n

And what you did above was correct, starting with (x + 1).<\/b><\/p>\n

(1) For the second term: add 1 to (x + 1)<\/b><\/p>\n

(2) For the 3rd term: add 1 again, and so on.<\/b><\/p>\n

(3) As I have hinted several times, there is not (necessarily) 5 terms<\/b><\/p>\n

(4) What is the last term?<\/b><\/p>\n<\/blockquote>\n

Hi again sir<\/p>\n

Thanks for replying back sir \ud83d\ude42<\/p>\n

But I thought the number above sigma indicates the number of terms (so in the case of <\/p>\n

\"\\sum_{a=x+1}^{5}{(a)}, <\/p>\n

there should be 5 terms in the sequence shouldn't there???)<\/p>\n

The textbook I have says this and also says if its infinity instead of 5, for example, then the sequence goes on and on. But the number above sigma essentially means there's a restriction to the number of terms in that sequence doesn't it??<\/p>\n

But even then sir, if<\/p>\n

\"\\sum_{a=x+1}^{5}{(a)}<\/p>\n

then wouldn't the second term be a + (x+1), like ...<\/p>\n

\"\\sum_{a=x+1}^{5}{(a)}
\n ... since we're adding on (x+1) because a = (x+1)??<\/p>\n

Sorry sir, but this is the first time I've come across an expression of this kind and I'm sure it's covered in the pure algebra units after (actually I just checked in the books and it isn't)!<\/p>\n

Sorry sir, but could you please explain what is meant by the part \"a=x+1\" and if you do substitute this into a<\/b>, do you then always (including for all other related expressions under sigma)<\/b> add 1 all the time??<\/p>\n

The only thing I do have in the textbooks similar to this is this ...<\/p>\n

\"textbook<\/p>\n

<\/i>And still, I don't know what's going on.<\/p>\n

Thanks again sir, and I'm really sorry I hope you can understand this is the first time I've come across something like this. The textbook has concentrated on an isolated value of a throughout.<\/p>\n

Thanks again sir \ud83d\ude42<\/p>\n

Yousuf.<\/i><\/p>\n

\n

No need to apologise! You are right - this is the first time you are coming across this.<\/b><\/p>\n

But please realise - in the \"real world\" we come across novel problems and what we need to do is to apply what we already know.<\/b><\/p>\n

That's what they are trying to do here - get you to apply what you know in a novel situation.<\/b><\/p>\n

Your answer above is getting closer, but still not quite. You have added (x + 1), each time, not \"1\" as it should be.<\/b><\/p>\n

I can understand why that textbook example didn't help at all!! Yet another case where they just assume you can figure it out.<\/b><\/p>\n

OK - back to the problem.<\/b><\/p>\n

It will only be 5 terms if you have a simple case like the one I gave you earlier. Only one variable is involved (a), and so there are 5 terms.<\/b><\/p>\n

\"\\sum_{a=1}^{5}a=1+2+3+4+5\"<\/p>\n

In summary, write the first number given at the bottom, add one to it then add that to the first number given, then add 1 again, add that to the other 2 already there.<\/b><\/p>\n

That's our process, whether we know an actual value for the first term or not.<\/b><\/p>\n

Question - is there an \"a\" in my expansion above?<\/b><\/p>\n

Now, try to expand these ones. I hope they lead you to enlightenment!<\/b><\/p>\n

\"\\sum_{a=2}^{5}a\"<\/p>\n

\"\\sum_{a=4}^{5}a\"\n <\/p>\n

\"\\sum_{a=-1}^{5}a\"<\/p>\n

\"\\sum_{a=0}^{5}a\"\n <\/p>\n

Were there 5 terms for each of the above? Did \"a\" appear in any of your answers?<\/b><\/p>\n

\"\\sum_{a=t}^{5}a\"<\/p>\n

\"\\sum_{a=x}^{5}a\"<\/p>\n

The above 2 answers should be almost identical, except for the letters you use. HINT: There will be no \"a\" in the answer.<\/strong><\/p>\n

Now try this one again:<\/strong><\/p>\n

\"\\sum_{a=x+1}^{5}a\"
\n <\/i><\/p>\n<\/blockquote>\n

Hi sir<\/p>\n

Thanks for reading this, and I'm sorry for not having replied back sooner.<\/p>\n

Right, I'll give each one a go...<\/p>\n

\"\\sum_{a=2}^{5}a=3+4+5=12\" <\/p>\n

substituting values from a = 2 to a = 5 <\/b><\/p>\n

\n

Correct - finishes when a = 5. There are 3 terms only in the series (not 5).<\/b><\/p>\n<\/blockquote>\n

\"\\sum_{a=4}^{5}a=4+5=9\" <\/p>\n

substituting values from a = 4 to a = 5<\/p>\n<\/p>\n

\n

Correct - finishes when a = 5. There are 2 terms only in the series (not 5).<\/b>\n<\/p>\n<\/blockquote>\n

\"\\sum_{a=-1}^{5}a=-1+0+1+2+3+4+5=14\" <\/p>\n

substituing values from a = -1 to a = 5<\/p>\n<\/p>\n

\n

Correct - finishes when a = 5. There are 7 terms in the series (not 5).<\/b>\n<\/p>\n<\/blockquote>\n

\"\\sum_{a=0}^{5}a=0+1+2+3+4+5=15\" <\/p>\n

substituting values from a = 0 to a = 5<\/p>\n

\n

Correct - finishes when a = 5. There are 6 terms in the series (not 5)<\/b><\/p>\n<\/blockquote>\n

\"math\"<\/p>\n

...subbing values from a = t to a = t + 4<\/p>\n

The last term has to be [t+4] because if a = 1, then the last term would be 5, in that [4+1] = [5].<\/p>\n

\n

NOT correct - you have not finished with a = 5. Why are there 5 terms in your series? I keep hinting there will NOT be 5 terms in the series!! (The start is correct)<\/b> <\/p>\n<\/blockquote>\n

\"math\"<\/p>\n

... subbing values from a = x to a = x+4<\/p>\n

The same condition would apply here also as said before.<\/p>\n

\n

NOT correct - you have not finished with a = 5, again. Why are there 5 terms in your series? (The start is correct)<\/b><\/p>\n<\/blockquote>\n

\"\\sum_{a=x+1}^{5}a\"<\/i><\/p>\n

The expression itself is saying that the first number, a, will be such that it will be +1 that of the first term, namely 1. <\/p>\n

\n

[NO - this is not what it is saying. Look at your correct patterns above!] <\/b><\/p>\n<\/blockquote>\n

For this one, what I plan to do is to start with the number 2 since a = x + 1, then the first term would be 2 as a = (1) + 1 = 2. The last term would then be 5 since a<\/i> requires the sequence of natural numbers, therefore ...<\/p>\n

\"\\sum_{a=x+1}^{5}a=2+3+4+5=14\"<\/p>\n

\n

NOT correct - you have not started correctly, using the correct pattern from above. But yay - we don't have 5 terms in the series!!<\/strong><\/p>\n<\/blockquote>\n

Did I do those OK sir?<\/p>\n

Thanks so much sir \ud83d\ude42<\/p>\n

Yousuf.<\/p>\n

 <\/p>\n

Hi again sir<\/p>\n

I'll give it another go...<\/p>\n

\"\\sum_{a=2}^{5}a=3+4+5=12\"<\/p>\n

In this example, what I did was that I substituted a = 2 as the first term, and then continued using the natural numbers until I reached a = 5.<\/p>\n

In the case of ...<\/p>\n

\"\\sum_{a=t}^{5}a\"<\/p>\n

... what I think it is saying is that the first term is a = t and the series will end when we reach a = 5.<\/p>\n

\n

But you didn't start with 2 above! What you wrote above is correct, but none of the following expansions is correct.<\/b><\/p>\n<\/blockquote>\n

(I know this is going to be wrong sir) Does that mean it will appear as ...<\/p>\n

\"\\sum_{a=t}^{5}a=t+[t+1]+[t+2]+[t+3]+[t+4]+[t+5]=5t+15=5(t+3)\"<\/p>\n

or would it be ...<\/p>\n

\"\\sum_{a=t}^{5}a=t+5\" <\/p>\n

since the first term is a = t and the last term is a = 5???<\/p>\n

Also, would the next one be...<\/p>\n

\"\\sum_{a=x}^{5}a=x+++++=5x+15=5(x+3)\"<\/p>\n

or would it be ...<\/p>\n

\"\\sum_{a=x}^{5}a=x+5\" (since the first term is a = x and the last term is a = 5???)<\/p>\n

OK since that is that done (hopefully right)...<\/p>\n

\"\\sum_{a=x+1}^{5}a\"<\/p>\n

IN all honestly sir, I still don't know what it's asking. Is this going to be an algebraic sequence if it's not numeric?<\/p>\n

Because I don't understand that if a = x+1, then what will this number be? I first thought it'd be 2, but if it's algebraic, then wouldn't it be (x+1).<\/p>\n

If it's (x+1), then shouldn't the next terms be ...<\/p>\n

(x+1) + 1<\/p>\n

(x+1) + 2<\/p>\n

(x+1) + 3<\/p>\n

(x+1) + 4<\/p>\n

(x+1) + 5\n<\/p>\n

So<\/p>\n

\"math\"<\/p>\n

I know it's going to be wrong sir, but I honestly don't have any other clue what else it could be?<\/p>\n

Thanks again sir \ud83d\ude42<\/p>\n

Yousuf.<\/p>\n

\n

You are so close, but you can't seem to accept the idea of a variable number of terms. You also need to finish with the number at the top.<\/p>\n

OK - try these. (And please, DON'T write out every term!!, and don't worry about finding a final answer as a single number. We are only interested in the expansion.)<\/p>\n

\n \"\\sum_{a=0}^{3000}{a<\/p>\n

\n \"\\sum_{a=20}^{10000}{a}
\n <\/b><\/p>\n

\"\\sum_{a=20}^{n}{a}<\/p>\n<\/blockquote>\n

Hi again sir<\/p>\n

I still don't get them sir, the ones for<\/p>\n

\"\\sum_{a=t}^{5}a\" <\/p>\n

(and the one with x) and also the one with<\/p>\n

\"\\sum_{a=x+1}^{5}a\".<\/p>\n

The ones above should be ...<\/p>\n

\"\\sum_{a=0}^{3000}{a<\/p>\n

\"\\sum_{a=20}^{10000}{a}<\/p>\n

and also<\/p>\n

\n\"\\sum_{a=20}^{n}{a}<\/p>\n

... shouldn't they??<\/p>\n

Thanks again sir \ud83d\ude42<\/p>\n

Yousuf.<\/p>\n

\n

\n CORRECT!! All 3 of these are correct. Now we are getting somewhere.<\/p>\n

In the first 2 examples (up to 3000 and up to 10000) we know how many terms there are but write \"...\" to indicate the pattern keeps going UP TO the end number.<\/p>\n

In the 3rd one, it finishes with \"n\". We don't know how many terms there are, but the \"...\" just means keep going up to n. We don't know what n is - we need more information to find it.<\/p>\n

Now it's the same deal when we don't know what the STARTING value is. We just start with that value involving x (like you did successfully above), do a few terms, THEN PUT \"...\" to indicate missing terms, then FINISH WITH THE TOP NUMBER (which in most of the above examples was 5.)<\/p>\n

Now, would you like to try those expansions with x in them again?
\n <\/b><\/p>\n<\/blockquote>\n

Hi again sir<\/p>\n

Thanks for reading this and apologies for not having replied back sooner.<\/p>\n

Right, I may get this wrong again sir, but does the following expand to...<\/p>\n

\"\\sum_{a=t}^{5}a=t+.\\<\/p>\n

and<\/p>\n

\"\\sum_{a=x}^{5}a=x+\\<\/p>\n

Sorry sir, but I'm convinced the above are incorrect, but I've drawn from the previous principle.<\/p>\n

I'm not quite sure if there is a pattern involving the first number (a = t or a = x) so I've assumed we start off with the first number as t<\/i> or x<\/i> and proceed therewith to the final number in the sequence, in both cases 5.<\/p>\n

Thanks again sir for your kind help and support \ud83d\ude41<\/p>\n

Yousuf.<\/p>\n

\n

Hi again<\/p>\n

You've got the first and last terms CORRECT!<\/p>\n

Let's go back to this example. The proper expansion should have the first few terms and the last few terms, with \"...\" in the middle, like this:
\n <\/b><\/p>\n

\"\\sum_{a=0}^{3000}{a<\/p>\n

So try these again, but put in at least 3 terms before the \"...\" and 3 terms after.<\/b><\/p>\n

\"\\sum_{a=t}^{5}a=\"<\/p>\n<\/p>\n

\"\\sum_{a=x}^{5}a=\"<\/p>\n<\/blockquote>\n

Hi again sir<\/p>\n

Thanks for replying back sir.<\/p>\n

Oh right, so would they be ..<\/p>\n

\"\\sum_{a=t}^{5}a=(t+1)+(t+2)+(t+3)+<\/p>\n

\"\\sum_{a=x}^{5}a=(x+1)+(x+2)+(x+3)+<\/p>\n

Sir I know you might be thinking I've cheated above to not have put any values before 5 in each instance! But I'm not sure what goes after the \"+...+\". I was going to write t<\/i>+n<\/i> and x+<\/i>n where n<\/i> could be any number but haven't done so in knowing it's going to be wrong sir!!<\/p>\n

I know from the expressions that ...<\/p>\n