If x, 2x + 2, 3x + 3 are in G.P. then 4th term is A) -27 B) 13.5 C)

1.	If x, 2x + 2, 3x + 3 are in G.P. then 4th term is A) -27 B) 13.5 C) 27 D) -13.5
Answer» Correct option is (D) -13.5 Given that x, 2x+2, 3x+3 are in G.P. \(\therefore(2x+2)^2=x(3x+3)\) \(\Rightarrow4x^2+8x+4=3x^2+3x\) \(\Rightarrow x^2+5x+4=0\) \(\Rightarrow(x+1)(x+4)=0\) \(\Rightarrow x+1=0\;or\;x+4=0\) \(\Rightarrow x=-1\;or\;x=-4\) Case I :- If x = -1, then \(2x+2=-2+2=0\) \(3x+3=-3+3=0\) \(\because-1,0,0\) are not in G.P. but x, 2x+1, 3x+3 are not in G.P. which is contradiction. Case II :- If x = -4 Then \(2x+2=-8+2=-6\) & \(3x+3=-12+3=-9\) \(\because-4,-6,-9\) are in G.P. which is required. \(\therefore r=\frac{-6}{-4}=\frac32\) Next term \(=-9r=-9\times\frac32\) \(=\frac{-27}2=-13.5\) Hence, \(4^{th}\) term is -13.5 Correct option is D) -13.5

If x, 2x + 2, 3x + 3 are in G.P. then 4th term is A) -27 B) 13.5 C) 27 D) -13.5

Answer»

Correct option is (D) -13.5

Given that x, 2x+2, 3x+3 are in G.P.

\(\therefore(2x+2)^2=x(3x+3)\)

\(\Rightarrow4x^2+8x+4=3x^2+3x\)

\(\Rightarrow x^2+5x+4=0\)

\(\Rightarrow(x+1)(x+4)=0\)

\(\Rightarrow x+1=0\;or\;x+4=0\)

\(\Rightarrow x=-1\;or\;x=-4\)

Case I :-

If x = -1, then

\(2x+2=-2+2=0\)

\(3x+3=-3+3=0\)

\(\because-1,0,0\) are not in G.P.

but x, 2x+1, 3x+3 are not in G.P. which is contradiction.

Case II :-

If x = -4

Then \(2x+2=-8+2=-6\)

& \(3x+3=-12+3=-9\)

\(\because-4,-6,-9\) are in G.P. which is required.

\(\therefore r=\frac{-6}{-4}=\frac32\)

Next term \(=-9r=-9\times\frac32\)

\(=\frac{-27}2=-13.5\)

Hence, \(4^{th}\) term is -13.5

Correct option is D) -13.5