We use summation rules to simplify our work. Read summation rules carefully; otherwise, it sounds like double talk! Note that we consistently use k as the symbol for a constant.

Rule 1: The summation of the sums of two or more variables equals the sum of their summations

$\sum\limits_{i=1}^n(x_{i}+y_{i})=\sum\limits_{i=1}^nx_{i} + \sum\limits_{i=1}^ny_{i}$

Rule 2: The summation of a constant times the values of a variable is equal to the constant times the summation of the variable

$\sum\limits_{i=1}^nkx=k\sum\limits_{i=1}^nx$

Rule 3: The summation of a constant taken N times is the constant times N

$\sum\limits_{i=1}^nk=kn$

Rule 4: The summation of the values of a variable plus a constant is equal to the summation of the values of the variable plus N times the constant

$\sum\limits_{i=1}^n(k+x)=nk + \sum\limits_{i=1}^nx$

Rule 5: The summation of the values of a variable minus a Constant is equal to the summation of the values of the variable minus N times the constant

$\sum\limits_{i=1}^n(x-k)=\sum\limits_{i=1}^nx-nk$