Sequence and SeriesMCQPYQ July 21Question 1871 of 212
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The sum of three numbers in a geometric progression is 28\displaystyle 28. When 7\displaystyle 7, 2\displaystyle 2 and 1\displaystyle 1 are subtracted from the first, second and the third numbers respectively, then the resulting numbers are in arithmetic progression. What is the sum of squares of the original three numbers?

Options

A510\displaystyle 510
B456\displaystyle 456
C400\displaystyle 400
D336\displaystyle 336
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Correct Answer

✅ Option d — 336\displaystyle 336

All Options:

  • A510\displaystyle 510
  • B456\displaystyle 456
  • C400\displaystyle 400
  • D336\displaystyle 336

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Detailed Solution & Explanation

Let the three numbers be a,ar,ar2\displaystyle a, ar, ar^2.
Given:
a+ar+ar2=28  ⟹  a(1+r+r2)=28— (1)a + ar + ar^2 = 28 \implies a(1+r+r^2) = 28 \quad \text{--- (1)}

Subtracting 7,2,1\displaystyle 7, 2, 1 yields an A.P. series a−7,ar−2,ar2−1\displaystyle a-7, ar-2, ar^2-1:
2(ar−2)=(a−7)+(ar2−1)2(ar-2) = (a-7) + (ar^2-1)
2ar−4=a+ar2−82ar - 4 = a + ar^2 - 8
a(r2−2r+1)=4  ⟹  a(1−r)2=4— (2)a(r^2 - 2r + 1) = 4 \implies a(1-r)^2 = 4 \quad \text{--- (2)}

Dividing equation (1) by equation (2):
1+r+r2(1−r)2=284=7\frac{1+r+r^2}{(1-r)^2} = \frac{28}{4} = 7
1+r+r2=7(1−2r+r2)1+r+r^2 = 7(1-2r+r^2)
6r2−15r+6=0  ⟹  2r2−5r+2=06r^2 - 15r + 6 = 0 \implies 2r^2 - 5r + 2 = 0
(2r−1)(r−2)=0  ⟹  r=2 or r=12(2r-1)(r-2) = 0 \implies r = 2 \text{ or } r = \frac{1}{2}

If r=2\displaystyle r=2, from (1): 7a=28  ⟹  a=4\displaystyle 7a = 28 \implies a = 4. The numbers are 4,8,16\displaystyle 4, 8, 16.
Sum of squares of original numbers is:
42+82+162=16+64+256=3364^2 + 8^2 + 16^2 = 16 + 64 + 256 = 336
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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