I really don't 'belong' here, but I'd love to have this question answered for my granddaughter.... one of my teen grandkids asked me a question about the purpose of using 'complicated' equations when an answer could be determined with 'simple' multiplication.
As an uneducated man, the only answer I could come up with was it uses a lot fewer #s and space, so I decided to look for a better response. I found this site and I found the following equation that was posted in a previous question and it's a perfect example of what she was asking about.
Why is it better to use the equation than to use multiplication?
Thanks for your help, she's a smart kid, I want to do anything that will add to her knowledge.
John Savarese,
Waltham, MA

On the first Sunday of 2003, Rizzo and Frenchie start a chain letter, each of them sending five letters (to ten different friends between them). Each person receiving the letter is to send copies to five new people on the Sunday following the letter’s arrival. After the first seven Sundays have passed, what is the total number of chain letters that have been mailed? How many were mailed on the last three Sundays?
The way I solved it was to add them together to find the number of letters that have been mailed.
a) $2 \cdot 5 + 2 \cdot 5^2 + 2 \cdot 5^3 + 2 \cdot 5^4 + 2 \cdot 5^5 + 2 \cdot 5^6 + 2 \cdot 5^7 = 195310$
b) $2 \cdot 5^5 + 2 \cdot 5^6 + 2 \cdot 5^7 = 193750$
 A: Hmm, I'm not sure what your granddaughter means by "complicated equations". If she's referring to the $\sum$ symbol, used for indicating the sum of a bunch of numbers, here are a few advantages:

*

*We sometimes want to add up a million terms, and the $\sum$ notation keeps the formula small no matter how many terms we want.


*We also might want to consider taking a varying number of terms. "Sigma notation", which is the name for all that $\sum$  business, lets us do that too.
If she's irritated by having to introduce the $i$ variable, figure out the lower and upper bounds, and then write the term as $5^i$, when using "..." seems clear enough to her, then tell her

*

*The sigma notation makes it absolutely clear what you are doing, whereas the "..." can sometimes be interpreted differently by different people.


*And one huge advantage is that the sigma notation makes it so explicit that we can get a computer to calculate it for us! That's not such a big deal when there are only five or six terms, but when there are a million of them it is. Despite how good computers have gotten at understanding regular language lately, they're still not very good at dealing with "....". So if they're going to handle our money or our medicines, it's still a good idea to give them their instructions in a format that can't be misinterpreted.
Hope that helps, and "Hello" from the Fenway! What a great grandad to help his grandkids like this.
