If \(x, 2x+2, 3x+3\) are in G.P., then the 4th term is:
Answer: A
If the terms are in G.P., the ratio of consecutive terms is constant. So, \(\frac{2x+2}{x} = \frac{3x+3}{2x+2}\). \((2x+2)^2 = x(3x+3) \Rightarrow 4x^2+8x+4 = 3x^2+3x \Rightarrow x^2+5x+4=0 \Rightarrow (x+1)(x+4)=0\). So \(x=-1\) or \(x=-4\). If x=-1, terms are -1, 0, 0 (not a G.P.). If x=-4, terms are -4, -6, -9. This is a G.P. with \(r = -6/-4 = 3/2\). The 4th term is \(-9 \times (3/2) = -27/2 = -13.5\).