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!!!!!