Given the sequence 5; 12 ; 21; 32; .... a determine the formular for the nth term of thw sequance . 2 determine between which two consecutive terms in the sequance the difference will equal 245 sketch the graph to represent the second difference
second difference is constant so it could defined by quadratic formula, u(x) = ax^2 + bx + c because 2a = Second difference = 2 so a = 1 then u(x) = x^2 + bx + c
build equations from known values to find b and c for x=1, u(1) = 5 = 1^2 + b(1) + c b + c = 4 ...(1)
x=2, u(2) = 12 = 2^2 + b(2) + c 2b + c = 8 ... (2)
solve b = 4, c =0 so u(x) = x^2 + 4x ... formula for n term (i use x not n here)
difference between consecutive term is 245, we have 245 = u(x+1) - u(x) 245 = (x+1)^2 + 4(x+1) - (x^2 + 4x) 245 = x^2 + 2x + 1 + 4x + 4 - x^2 - 4x 245 = 2x + 5 x = 120 it's between term 120 and 121
