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

First difference

12-5 = 7

21-12 = 9

32-21 = 11

Second difference

9-11 = 2

11-9 = 2

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

graph is y=2