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In the section on arithmetic series, we talked about how to find the sum of a specified number of terms in an arithmetic sequence. Essentially, we learned how to go quickly from a list of numbers to an overall sum:
But writing out n terms can get messy. Further, if we have limited information on the sequence itself or the number of terms, it can be tedious or tricky to use algebra on the formula for Sn in order to discover these things and, if we want to alter the sequence slightly, annoying to go through the whole process over again.
Let's work through an example to see what I mean. Consider the series:
The terms form an arithmetic sequence with first term u1=−2 and common difference d=3. We can find the number of terms by solving Sn=245:
Since n>0 in the context of this problem,
Already, that was a lot of work for a question that seemed relatively simple. Now what if I change the sequence somehow, multiplying each term by a factor of x and adding y, and I give you the sums of some number of terms but I don't tell you what x and y are?
How do we even begin to answer this? It seems like a very complicated question when we ask it like this. But if we use "summation notation," we can both ask and answer it in a more straightforward manner.
Let's start with our original series and sum, S14=−2+1+⋯+37. We can condense this sum and write it algebraically using the so-called ∑ ("sigma") notation:
Discussion
Explain the different pieces in the expression n=1∑14(3n−5).
What does the "n=1" do?
How about the "14"?
What about the 3n−5?
The notation
is read “the sum, as n runs from 1 to 15, of the terms 3n−5.” In detail:
The symbol ∑ means “add up all the following terms.”
The subscript n=1 tells us to start with n=1. In other words, the first term is
The superscript 14 tells us to stop at n=14. The last term is
The expression 3n−5 is the general term: for each integer n between 1 and 14, we compute 3n−5 and then add all those values together.
So we form the sequence of 14 terms
and sum them.
Understanding summation notation
As a shortcut for writing out long sums, we can use the symbol ∑ with the following "settings":
Here n is called the index, but other letters can also be used in place of n.
Checkpoint
Write out n=1∑6(3+2n) explicitly as a sum of terms.
Select the correct option
Exercise
Given that n=1∑15(5+kn)=−105, find the value of k.
Select the correct option
Discussion
What would be the value of n=1∑223, and why?
To compute
note that this is just adding 3 to itself 22 times:
Hence
Equivalently, viewing it as an arithmetic series with first and last term both 3 and 22 terms,
so the value of the sum is 66.
This property can be generalized to the sum of any constant:
Sum of a constant
All this is saying is that if I add up the same number, c, n times, I just get n times c -- which is basically just restating the definition of multiplication.
Checkpoint
Given that n=1∑k5=105, find k.
Select the correct option
Discussion
It is given that n=1∑8n2=204.
What is n=1∑82n2?
How about n=1∑83n2?
Can you explain why?
Write out the first few terms of n=1∑82n2:
Each term has a factor 2, so we can “pull” it in front of the parentheses:
Similarly,
Basically, if you have a sum where each term is scaled by some number c, instead of multiplying each term by c and then adding them all together, you can add together the unscaled terms and then multiply that sum by c, i.e.,
Sums with scalar multiples
Checkpoint
Given that n=1∑kan=24 and n=1∑kcan=168, what is the value of c?
Select the correct option
Discussion
It is given that n=1∑4an=44 and n=1∑4bn=23. Evaluate the sum n=1∑4(an+bn).
First, write out every term:
Next, notice we can collect all the a–terms together and all the b–terms together:
Rewriting each parenthesis in sigma notation gives
Finally substitute the given values:
Sum of a sum
Essentially, any ∑ of a sum can be broken into two ∑ 's.
Or vice versa! If two sums have the same start and stop index (eg k=1 up to n), they can be merged.
Checkpoint
It is given that n=1∑15an=130, a1 = 7 and a15 = 13.
Determine the value of n=2∑14an.
Select the correct option
How would you go about figuring out how many red dots there are on the diagram to the right?
First, notice the "obvious" fact that the total number of dots is the number of blue dots plus the number of red dots.
That means the number of red dots is the difference between the total dots and the number of blue dots.
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Now, notice that the rows of the triangle form an arithmetic sequence with a1=1 and a common difference of 2.
Then, the number of blue dots may be represented as n=1∑4(−1+2n), which is equal to 16, and the number of total dots may be represented as n=1∑8(−1+2n), which is equal to 64.
Finally, by subtracting the blue dots from the total dots, we see that there are 48 red dots.
The key intuition we used to figure out this is answer can be expressed algebraically as
All this is really saying is that we can split a ∑ anywhere in the middle.
Splitting a sum in Σ form
For any series of the form k=1∑nak and any integer m between 1 and n, we can split the series at the index m:
Discussion
Remember from earlier that n=1∑kn2=6k(k+1)(k+2).
What is the value of n=1∑10n2?
Can you use that answer, together with the general formula for n=1∑kn2, to find n=5∑10n2?
First, recall the formula for the sum of squares:
Compute the sum of the first ten squares:
Notice that writing out all ten terms gives
Here the first four squares 12,⋯,42 and the last six squares 52,⋯,102 together make up the full list of ten. Thus
Compute the sum of the first four squares:
Finally, subtract:
Exercise
It is given that k=1∑8ak=30, k=1∑4ak=12, and k=5∑8bk=5.
Find k=5∑8(2ak−bk)
Select the correct option
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