Question

find the value of x if 2 X + 1 ,X square + X + 1, 3 x square - 3 x + 3 are consecutive terms of an ap

Answer 1

Given, 2x+1, x^2 +x+1 and 3x^2 - 3x+ 3 are the consecutive terms of an A.P.
As the terms are in ap, therefore common difference between the given consecutive terms will be equal.
So,
(x + x + 1) - (2x + 1) = (3x - 3x+ 3) - (x + x + 1)
⇒ 2(x + x + 1) = (3x - 3x+ 3) + (2x + 1)
⇒ 2x + 2x + 2 = 3x - x + 4
⇒ 0 = x - 3x + 2
⇒ x - 3x + 2 = 0
⇒ x - 2x - x + 2 = 0
⇒ x(x - 2) -1(x - 2) = 0
⇒ (x - 2)(x -1) = 0
⇒ x = 2 or x = 1

Answer 2

2x + 1 =2x + x +1 = 3x -3x +3
= 3x +5
= x =5-3
= x=2