Writing Formula for Sigma Notation Sum [k=1 to n] (3- k) for given n-values

[lroscios8232]

Hello everyone, I've just started integrals.

I'm working on some homework problems and I need help explaining this one. This question is included in a section with instructions:

"Find a formula for the sum, then use the formula to calculate each sum for n = 100, n = 500, and n = 1000." (I will be using E in place of sigma).

n
E (3-k)
k=1

If someone could, please walk through the steps and solve this so I can know how to encounter this problem from now on?

 

[Deleted member 4993]

Hello everyone, I've just started integrals.

I'm working on some homework problems and I need help explaining this one. This question is included in a section with instructions:

"Find a formula for the sum, then use the formula to calculate each sum for n = 100, n = 500, and n = 1000." (I will be using E in place of sigma).

n
E (3-k)
k=1

If someone could, please walk through the steps and solve this so I can know how to encounter this problem from now on?

What are your thoughts?

Please share your work with us ...even if you know it is wrong.

If you are stuck at the beginning tell us and we'll start with the definitions.

You need to read the rules of this forum. Please read the post titled "Read before Posting" at the following URL:

http://www.freemathhelp.com/forum/announcement.php?f=33

 

[JeffM]

First, there are some rules for handling expressions that use sigma-notation. Some are shown at the end of the following site

http://www.mathcentre.ac.uk/resources/uploaded/mc-ty-sigma-2009-1.pdf

Second, there are methods to prove logically that formulas involving sigma-notation are correct.

But third, discovering the formulas is an art, not a science.

To get you started, I am going to tell you one rule that is pertinent to your problem, and show you one common way to discover a rule.

The pertinent rule is basic and pretty obvious once you remember that the order in which you add makes no difference. The rule is

\(\displaystyle \displaystyle \left ( \sum_{j=1}^n f(j) + g(j) \right ) \equiv \left ( \sum_{j=1}^n f(j) \right ) + \left ( \sum_{j=1}^n g(j) \right ).\)

This rules lets you break down a complex sum into simpler sums.

When it comes to discovering formulas, sometimes you can do it by pure reasoning, but usually it takes detective work. Let's try an example.

Is there a formula for the sum of the SQUARES of the first n ODD numbers?

\(\displaystyle n = 1 \implies \displaystyle \sum_{j=1}^n (2j - 1)^2 = 1.\)

\(\displaystyle n = 2 \implies \displaystyle \sum_{j=1}^n (2j - 1)^2 = 1 + 3 = 4.\)

\(\displaystyle n = 3 \implies \displaystyle \sum_{j=1}^n (2j - 1)^2 = 1 + 3 + 5 = 9.\)

\(\displaystyle n = 4 \implies \displaystyle \sum_{j=1}^n (2j - 1)^2 = 1 + 3 + 5 + 7 = 16.\)

Do you see a pattern that leads to a formula? How about

\(\displaystyle \displaystyle \sum_{j=1}^n (2j - 1)^2 = j^2.\)

 

[Dr.Peterson]

One reason we want you to show some sort of work is that without it, we don't know what methods you are learning. JeffM's answer might be exactly what you need, or it may totally baffle you.

If you can't start this problem on your own, can you at least show us an example of the way your textbook or teacher showed you to find sums like this?

My guess is that you may have seen formulas for sums like \(\displaystyle \sum_{k=1}^{n}1\) and \(\displaystyle \sum_{k=1}^{n}k\), which you can combine (using the rules JeffM referred you to) to get your sum. Is that true? If so, give it a try.