Question 210252
the differences between consecutive Q terms are ___ 9, 13, 17, 21, 25


the difference between consecutive terms of the 1st set of differences is equal to 4 and is constant
___ this means that the conjecture is a second order expression (2nd differences constant)


Q(n) = an^2 + bn + c


Q(1) = a(1^2) + b(1) + c ___ 3 = a + b + c ___ 3 - a - b = c


Q(2) = a(2^2) + b(2) + c ___ 12 = 4a + 2b + c
substituting ___ 12 = 4a + 2b + 3 - a - b ___ 9 = 3a + b


Q(3) = a(3^2) + b(3) + c ___ 25 = 9a + 3b + c
substituting ___ 25 = 9a + 3b + 3 - a - b ___ 22 = 8a + 2b ___ 11 = 4a + b


subtracting equations ___ 2 = a


substituting ___ 11 = 4(2) + b ___ 3 = b


substituting ___ 3 - 2 - 3 = c ___ -2 = c


Q(n) = 2n^2 + 3n - 2


Q(100) = 2(100^2) + 3(100) - 2 = 20298