Sequence and SeriesPYQ Sept 25Question 4429 of 150
All Questions

If the numbers x,2x+2\displaystyle x, 2x + 2 and 3x+3\displaystyle 3x + 3 are in the geometric progression, then the fourth term of the progression is

Options

A27
B-27
C13.5
D-13.5
For any discrepancies in this question, email contact@cadada.in

Correct Answer

✅ Option d — -13.5

All Options:

  • A27
  • B-27
  • C13.5
  • D-13.5

Detailed Solution & Explanation

Let the three terms in geometric progression (GP) be: t1=x,t2=2x+2,t3=3x+3t_1 = x, \quad t_2 = 2x+2, \quad t_3 = 3x+3
For these terms to be in GP, the ratio of consecutive terms must be equal: t2t1=t3t2  ⟹  t22=t1⋅t3\frac{t_2}{t_1} = \frac{t_3}{t_2} \implies t_2^2 = t_1 \cdot t_3 Substitute the expressions: (2x+2)2=x(3x+3)(2x+2)^2 = x(3x+3) 4(x+1)2=3x(x+1)4(x+1)^2 = 3x(x+1)
We have two cases: **Case 1:** x+1=0  ⟹  x=−1\displaystyle x + 1 = 0 \implies x = -1 If x=−1\displaystyle x = -1, the terms are −1,0,0\displaystyle -1, 0, 0, which is not a valid geometric progression because the common ratio would be undefined.
**Case 2:** x+1≠0\displaystyle x + 1 \ne 0 Dividing both sides by (x+1)\displaystyle (x+1): 4(x+1)=3x4(x+1) = 3x 4x+4=3x  ⟹  x=−44x + 4 = 3x \implies x = -4
Let's find the first three terms of this GP with x=−4\displaystyle x = -4: t1=−4t_1 = -4 t2=2(−4)+2=−6t_2 = 2(-4) + 2 = -6 t3=3(−4)+3=−9t_3 = 3(-4) + 3 = -9
The common ratio (r\displaystyle r) is: r=t2t1=−6−4=1.5r = \frac{t_2}{t_1} = \frac{-6}{-4} = 1.5
The fourth term (t4\displaystyle t_4) is: t4=t3⋅r=−9⋅1.5=−13.5t_4 = t_3 \cdot r = -9 \cdot 1.5 = -13.5
Hence, **Option D** is the correct answer.

About This Chapter: Sequence and Series

Paper

Paper 3: Quantitative Aptitude

Weightage

4-6 Marks

Key Topics

Arithmetic & Geometric Progressions

This chapter covers Arithmetic Progressions (AP) and Geometric Progressions (GP). Students learn how to find the nth term, sum of n terms, arithmetic/geometric means, and sum to infinity of a GP.

View Official ICAI Syllabus

Exam Strategy Tip

For complex 'sum of series' questions, a great hack is to substitute n = 1 and n = 2 into the question and the options to see which one matches, completely bypassing the formula.

More Questions from Sequence and Series

Sum of progression (a+b)\displaystyle (a+b), (a−b)\displaystyle (a-b), (a−3b)...n\displaystyle (a-3b)...n term is

Find the sum of first twenty-five terms of A.P. series whose nth\displaystyle n^{th} term is (n+25)\displaystyle \left(n+\frac{2}{5}\right)

The first and fifth term of an A.P. of 40\displaystyle 40 terms are −29\displaystyle -29 and −15\displaystyle -15 respectively. Find the sum of all positive terms of this A.P

If the common difference of an AP equals to the first term, then the ratio of its mth\displaystyle m^{th} term and nth\displaystyle n^{th} term is:

Find the value of 1+2+3+...+105\displaystyle 1+2+3+...+105

The first and last terms of an arithmetic progression are 5\displaystyle 5 and 905\displaystyle 905. Sum of the terms is 45,955\displaystyle 45,955. The number of terms is

Ready to Master Sequence and Series?

Practice all 150 questions with instant feedback, earn XP, track your streaks, and ace your CA Foundation exam.

Start Practicing — It's Free