Σ |

a≤i<B |

b | |

Σ | . |

i=a |

One nice consequence is that
the number of terms in the sum is simply *B* − *a*
rather than *b* − *a* + 1.
Perhaps more importantly, we have this theorem:

Σ | + | Σ | = | Σ | , |

A≤i<B |
B≤i<C |
A≤i<C |

b |
c |
c | |||

Σ | + | Σ | = | Σ | . |

i=a | i=b+1 | i=a |

The formulas for summing cubic polynomials also tend to be slightly simpler. With the traditional numbering, we have these (from the textbook):

- The sum, from
*i*= 0 to*i*=*b*, of a constant*c*is*c*(*b*+ 1). - The sum, from
*i*= 0 to*i*=*b*, of*i*itself is*b*(*b*+ 1)/2. - The sum, from
*i*= 0 to*i*=*b*, of*i*^{2}is*b*(*b*+ 1)(2*b*+ 1)/6. - The sum, from
*i*= 0 to*i*=*b*, of*i*^{3}is*b*^{2}(*b*+ 1)^{2}/4.

- The sum, over 0 ≤
*i*<*B*, of a constant*c*is*c**B*. - The sum, over 0 ≤
*i*<*B*, of*i*itself is*B*(*B*− 1)/2. - The sum, over 0 ≤
*i*<*B*, of*i*^{2}is*B*(*B*− 1)(2*B*− 1)/6. - The sum, over 0 ≤
*i*<*B*, of*i*^{3}is*B*^{2}(*B*− 1)^{2}/4.

The upshot of all of this is that,
when you see (for example) a sum as *i* runs from 2 to **5**,
you might want to think of it as
a sum over 2 ≤ *i*< **6** instead.

It's also handy to have more general formulas for summing cubic polynomials,
starting at an arbitrary place rather than at *i* = 0.
With the traditional numbering, we have these:

- The sum, from
*i*=*a*to*i*=*b*, of a constant*c*is*c*(*b*−*a*+ 1). - The sum, from
*i*=*a*to*i*=*b*, of*i*itself is (*a*+*b*)(*b*−*a*+ 1)/2. - The sum, from
*i*=*a*to*i*=*b*, of*i*^{2}is (2*a*^{2}+ 2*a**b*+ 2*b*^{2}−*a*+*b*)(*b*−*a*+ 1)/6. - The sum, from
*i*=*a*to*i*=*b*, of*i*^{3}is (*a*^{2}+*b*^{2}−*a*+*b*)(*a*+*b*)(*b*−*a*+ 1)/4.

- The sum, over
*A*≤*i*<*B*, of a constant*c*is*c*(*B*−*A*). - The sum, over
*A*≤*i*<*B*, of*i*itself is (*B*−*A*)(*A*+*B*− 1)/2. - The sum, over
*A*≤*i*<*B*, of*i*^{2}is (*B*−*A*)(2*A*^{2}+ 2*A**B*+ 2*B*^{2}− 3*A*− 3*B*+ 1)/6. - The sum, over
*A*≤*i*<*B*, of*i*^{3}is (*B*−*A*)(*A*+*B*− 1)(*A*^{2}+*B*^{2}−*A*−*B*)/4.

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