



Math is
like a foreign language, the symbols all stand for words and ideas.



The S
symbol, Greek letter capital Sigma, has a fairly involved meaning. 


It tells
us to sum up a bunch of terms. 


It is
kind of like a little computer program. 





How many
terms are we talkin' about? 





That bit
of info comes from other symbols that are above and below the S. 


You might
see something like: 








Maybe
instead of the X you will see a Y or a k or any old letter below the
S. 


The
important thing is that if you count from the bottom number (here a
1), 


to the
top number (here a 5) you get the number of terms we add together. 





1,
2, 3, 4, 5 (5 terms)






If we
had: 





We would
also have 5 terms. 


103,
104, 105, 106, 107 (5 terms)






Or: 





Which
would be: 


4,
3, 2, 1, 0 (5 terms)






WHY THIS
SILLY NOTATION? 


If you
wanted 5 terms, why not just write "5 terms" under the S
or something? 





The
reason is that those particular numbers: 


1, 2, 3, 4, 5 or 103, 104,
105, 106, 107 


might be
needed somewhere in the problem. 


Where we
wrote (stuff), 


we might
have something like: 


3^{X}



This
makes the whole expression: 





Did you
see that the letter under the S



matches the variable in the expression to the right? 


This
little group of symbols means that we have 5 terms. 


Each of
the terms is the number 3, raised to a power. 


The power
is the sequence of numbers. 





This
particular whole sum is: 


3^{1 }+
3^{2 }+ 3^{3 }+ 3^{4 }+ 3^{5 }= 363 





Which is
a whole lot different from: 





Which is: 


3^{103
}+ 3^{104 }+ 3^{105 }+ 3^{106 }+ 3^{107
}»
1.684 x 10^{51}






Sometimes
there is no variable in the statement. 


We might
have: 





Which is
just: 


4 + 4 + 4 + 4 +
4 + 4^{ }=
24



We have a
total of 6 fours here. 


That's
the same as 6 x 4. 


so we can
say that if we're "summing" a term with no variable 


and the
number under the S
is a 1 


the
answer is the number on top times the constant. 


In math
talk, that is: 





So: 








If we
have something like: 








We get: 


3(1) + 3(2) +
3(3) + 3(4) + 3(5)^{ }=
45






Remember
when we did limits, and derivatives, and integrals? 


The rules
let us move constants to the left and deal with them later. 


We can do
that here too. 


So: 








Also,
when we worked with limits, and derivatives, and integrals 


and had a
bunch of terms, like: 


3X^{2}
+ 4X  5






We could
work with each term by itself. 


We can do
that here too. 


So: 








Even so,
calculating this out would take a long time. 


Just the
first part would be: 


3(1^{2
}+ 2^{2 }+ 3^{2 }+ 4^{2 }+ 5^{2})






And these
things can get even more complex. 


How about
some shortcuts? 


SURE! 





If we
have X to the first power: 








If we
have X to the second power (X ^{2}): 








If we
have X to the third power (X ^{3}): 








If we
have some number to the X power (C ^{X}): 








copyright 2005 Bruce Kirkpatrick 
