Sequence and SeriesMCQPYQ Nov. 20Question 1828 of 213
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Three numbers in G.P. with their sum 130 and their product 27,000 are:

Select Your Answer (Keyboard: A, B, C, D)
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Correct Answer

✅ Option c — Both (a) & (b)

All Options:

  • A10, 30, 90
  • B90, 30, 10
  • CBoth (a) & (b)
  • D10, 20, 30

Detailed Solution & Explanation

Let the three numbers in G.P. be ar,a,ar\displaystyle ar\displaystyle \frac{a}{r}, a, ar.
Their product is $27,000\displaystyle :<br><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mfrac><mi>a</mi><mi>r</mi></mfrac><mo>⋅</mo><mi>a</mi><mo>⋅</mo><mi>a</mi><mi>r</mi><mo>=</mo><msup><mi>a</mi><mn>3</mn></msup><mo>=</mo><mn>27000</mn><mtext>  </mtext><mo>⟹</mo><mtext>  </mtext><mi>a</mi><mo>=</mo><mn>30</mn></mrow><annotation encoding="application/x-tex">ar\displaystyle \frac{a}{r} ⋅\displaystyle \cdot a ⋅\displaystyle \cdot ar = a^3 = 27000 \implies a = 30</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.7936em;vertical-align:-0.686em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.1076em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.02778em;">r</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">a</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">⋅</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.4445em;"></span><span class="mord mathnormal">a</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">⋅</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">a</span><span class="mord mathnormal" style="margin-right:0.02778em;">r</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8641em;"></span><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">3</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6684em;vertical-align:-0.024em;"></span><span class="mord">27000</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">⟹</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">a</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">30</span></span></span></span></span><br><br>Their sum is &#36;130:
30r+30+30r=130  ⟹  30(1r+r)=10030r+30+30r=130\displaystyle \frac{30}{r} + 30 + 30r = 130 \implies 30\left(1r+r\displaystyle \frac{1}{r} + r\right) = 100
1+r2r=103  ⟹  3r2−10r+3=01+r2r=103\displaystyle \frac{1+r^2}{r} = \frac{10}{3} \implies 3r^2 - 10r + 3 = 0
Factorizing:
(3r−1)(r−3)=0  ⟹  r=3 or r=13(3r-1)(r-3) = 0 \implies r = 3  or \displaystyle \text{ or } r = 13\displaystyle \frac{1}{3}

If r=3\displaystyle r = 3, the numbers are $10, 30, 90.<br>If\displaystyle .<br>Ifr = 13\displaystyle \frac{1}{3}$, the numbers are $90, 30, 10$.
Hence, **Option C** 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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