Question

if x,2x+2,3x+3 are geometric progression then,fourth term is​

Answer 1

Answer: -13.5

Step-by-step explanation:

Given that,

(x), (2x + 2), (3x + 3), is a Geometric Progression.

We are instructed to find the 4th term.

According to Geometric Progression, if a,b,c are in G.P., then:

$\implies \dfrac{b}{a} = \dfrac{c}{b}$

Similarly, for the following G.P. we get:

$\implies \dfrac{2x+2}{x} = \dfrac{3x+3}{2x+2}\\\\\\\text{Cross multiplying we get:}\\\\\\\implies (2x+2)(2x+2) = (3x+3)(x)\\\\\implies (4x^2 + 8x + 4) = (3x^2 + 3x)\\\\\implies 4x^2 - 3x^2 +8x - 3x +4 = 0\\\\\implies x^2 +5x +4 = 0\\\\\implies x^2 +4x+x+4=0\\\\\implies x(x+4) +1(x+4) = 0\\\\\implies (x+1)(x+4) = 0\\\\\implies \boxed{x = -1,-4}$

Case 1: Taking x = -1

We get the G.P. to be:

→ -1, 0, 0

Since the common difference is 0/1 = 0, we get the 4th term also as 0.

Case 2: Taking x = -4

We get the G.P. to be:

→ -4, -6, -9

Hence the common ratio is (-6/-4) = (3/2).

Now the 4th term of a G.P. is given as: ar³, where a is the first term and r is the common ratio.

Hence the 4th term of the G.P. is:

$\implies a_4 = -4 \times \dfrac{3^3}{2^3} = -4 \times \dfrac{27}{8}\\\\\\\implies \boxed{a_4 = \dfrac{-27}{2} = -13.5}$