Sequence and SeriesMCQMTP June 2023 Series IQuestion 1859 of 212
All Questions

Four Geometric Means between 4\displaystyle 4 and 972\displaystyle 972 are

Options

A12,30,100,324\displaystyle 12, 30, 100, 324
B12,24,108,320\displaystyle 12, 24, 108, 320
C12,36,108,324\displaystyle 12, 36, 108, 324
D12,36,108,324\displaystyle 12, 36, 108, 324
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Correct Answer

✅ Option d — 12,36,108,324\displaystyle 12, 36, 108, 324

All Options:

  • A12,30,100,324\displaystyle 12, 30, 100, 324
  • B12,24,108,320\displaystyle 12, 24, 108, 320
  • C12,36,108,324\displaystyle 12, 36, 108, 324
  • D12,36,108,324\displaystyle 12, 36, 108, 324

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Detailed Solution & Explanation

Let the four geometric means be G1,G2,G3,G4\displaystyle G_1, G_2, G_3, G_4.
Then, the sequence 4,G1,G2,G3,G4,972\displaystyle 4, G_1, G_2, G_3, G_4, 972 is in G.P.
Here:
First term a=4\displaystyle a = 4
Total number of terms N=6\displaystyle N = 6
Sixth term t6=972\displaystyle t_6 = 972.

Using the GP formula:
t6=a⋅r5t_6 = a \cdot r^5
972=4⋅r5972 = 4 \cdot r^5
r5=243  ⟹  r=3r^5 = 243 \implies r = 3

Now, calculate the four geometric means:
G1=a⋅r=4⋅3=12G_1 = a \cdot r = 4 \cdot 3 = 12
G2=a⋅r2=4⋅9=36G_2 = a \cdot r^2 = 4 \cdot 9 = 36
G3=a⋅r3=4⋅27=108G_3 = a \cdot r^3 = 4 \cdot 27 = 108
G4=a⋅r4=4⋅81=324G_4 = a \cdot r^4 = 4 \cdot 81 = 324
So the four geometric means are 12,36,108,324\displaystyle 12, 36, 108, 324.
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.

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