Consider the statement "2 is a factor of n2 + 3n."

if n = 1, the statement is "2 is a factor of "
if n = 2, the statement is "2 is a factor of "
if n = 3, the statement is "2 is a factor of "
if n = k + 1, the statement before the algebra is simplified is "2 is a factor of "
if n = k + 1, the statement after the algebra is simplified is "2 is a factor of "

4; 15; 28; (k + 1)2 + 3(k + 1); k2 + 5k + 8

4; 20; 28; (k + 1)2 + 3(k + 1); k2 + 5k + 7

4; 10; 18; (k + 1)2 + 3(k + 1); k2 + 5k + 4

The answer is the last one:

4 , 10 , 18 , (k + 1)² + 3(k + 1) and k² + 5k + 4

Step-by-step explanation:

∵ 2 is a factor of n² + 3n

∵ n = 1 ⇒ ∴ (1)² + 3(1) = 1 + 3 = 4 ⇒ 2 is a factor of 4

∵ n = 2 ⇒ ∴ (2)² + 3(2) = 4 + 6 = 10 ⇒ 2 is a factor of 10

∵ n = 3 ⇒ ∴ (3)² + 3(3) = 9 + 9 = 18 ⇒ 2 is a factor of 18

∵ n = k + 1 ⇒ ∴ (k + 1)² + 3(k + 1) ⇒ before the simplify

∵ n = k + 1 ⇒ ∴ k² + 2k + 1 + 3k + 3 = k² + 5k + 4 ⇒ after simplify

mrbilldoesmath!

answer: a parabola

steps:

(x-1)(x+4)   = x(x+4) - 1(x+4) = (x^2 + 4x) - (x+4)

                = x^2 +4x -x -4   = x^2 +3x -4

which is a parabola

mrb