<s>[INST] Find the sum of all odd numbers between two polynomials I was asked this question by someone I tutor and was stumped.
Find the sum of all odd numbers between $n^2 - 5n + 6$ and $n^2 + n$ for $n \ge 4.$
I wrote a few cases out and tried to find a pattern, but was unsuccessful.
Call polynomial 1, $p(n) = n^2 - 5n + 6,$ then $p(4)=2.$  Next, call polynomial 2, $q(n)=n^2 + n,$ then $q(4)=20.$  Then adding all the odd numbers between 2 and 20 gives the following sum:
$3+5+7+9+11+13+15+17+19= 99. \\$
$p(5)= 6$ and $q(5)=30.$ Then adding all the odd numbers between 6 and 30 gives the following sum:  $7+9+11+13+15+17+19+21+23+25+27+29=216 \\$
$p(6)=12$ and $q(6)= 42.$  Then adding all the odd numbers between 12 and 42 give the following sum:  $13+15+17+19+21+23+25+27+29+31+33+35+37+39+41 = 405.$
From here I do not see any apparent patters.  This problem was given in a Pre-Calculus course, so clearly only elementary methods are expected by the students.  
Any help or advice would be much appreciated.  Thank you!!!!!
 [/INST] Because $n^2+n$ is even we can write the odd numbers between the two polynomials to sum as:
$$n^2+n-1,n^2+n-3,\cdots,n^2+n-(6n-7)$$
and because the sum of odd numbers between $1$ and $2k+1$ is $(k+1)^2$ we have :
$$\sum_{i=0}^{3n-4}(n^2+n-(2i+1))=(3n-3)(n^2+n)-(3n-3)^2 $$
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