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The sum of first three terms of a G.P. is 212\displaystyle \frac{21}{2} and their product is 27. Which of the following is not a term of the G.P., if the numbers are positive?

Options

A3
B23\displaystyle \frac{2}{3}
C32\displaystyle \frac{3}{2}
D6
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Correct Answer

✅ Option b — 23\displaystyle \frac{2}{3}

All Options:

  • A3
  • B23\displaystyle \frac{2}{3}
  • C32\displaystyle \frac{3}{2}
  • D6

Detailed Solution & Explanation

Let the first three positive terms of the G.P. be ar\displaystyle \frac{a}{r}, a\displaystyle a, and ar\displaystyle ar.
According to the problem, the product of these three terms is 27\displaystyle 27:
(ar)×a×(ar)=27\left(\frac{a}{r}\right) \times a \times (ar) = 27
a3=27  ⟹  a=3a^3 = 27 \implies a = 3
The sum of these three terms is 212\displaystyle \frac{21}{2}:
3r+3+3r=212\frac{3}{r} + 3 + 3r = \frac{21}{2}
Divide the entire equation by 3\displaystyle 3:
1r+1+r=72\frac{1}{r} + 1 + r = \frac{7}{2}
1r+r=72−1=52\frac{1}{r} + r = \frac{7}{2} - 1 = \frac{5}{2}
Now form a quadratic equation by multiplying both sides by 2r\displaystyle 2r:
2(1+r2)=5r  ⟹  2r2−5r+2=02(1 + r^2) = 5r \implies 2r^2 - 5r + 2 = 0
Factorize the quadratic equation:
2r2−4r−r+2=02r^2 - 4r - r + 2 = 0
2r(r−2)−1(r−2)=02r(r - 2) - 1(r - 2) = 0
(2r−1)(r−2)=0  ⟹  r=2orr=12(2r - 1)(r - 2) = 0 \implies r = 2 \quad \text{or} \quad r = \frac{1}{2}
Since the terms must be positive, we check both cases:
- If r=2\displaystyle r = 2, the terms are: 32\displaystyle \frac{3}{2}, 3\displaystyle 3, 6\displaystyle 6
- If r=12\displaystyle r = \frac{1}{2}, the terms are: 6\displaystyle 6, 3\displaystyle 3, 32\displaystyle \frac{3}{2}
In either case, the terms of the G.P. are 32\displaystyle \frac{3}{2}, 3\displaystyle 3, and 6\displaystyle 6. Comparing this with the options, 23\displaystyle \frac{2}{3} is not a term of this G.P.
Hence, **Option B** 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.

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