Ratio, Proportion, Indices, LogarithmMCQMTP Nov 18Question 895 of 305
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If x=6+6+6+...\displaystyle x = \sqrt{6 + \sqrt{6 + \sqrt{6 + ...}}} then the value of x\displaystyle x is

Options

A3\displaystyle 3
B2\displaystyle 2
C1\displaystyle 1
D6\displaystyle 6
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Correct Answer

✅ Option a — 3\displaystyle 3

All Options:

  • A3\displaystyle 3
  • B2\displaystyle 2
  • C1\displaystyle 1
  • D6\displaystyle 6

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Detailed Solution & Explanation

Given the infinite nested radical:
x=6+6+6+...x = \sqrt{6 + \sqrt{6 + \sqrt{6 + ...}}}
We can rewrite the expression because the nested radical inside is also equal to x\displaystyle x:
x=6+xx = \sqrt{6 + x}
Since x\displaystyle x is a square root of positive numbers, x\displaystyle x must be positive (x>0\displaystyle x > 0). Squaring both sides of the equation:
x2=6+xx^2 = 6 + x
Rearrange the equation into a standard quadratic form:
x2−x−6=0x^2 - x - 6 = 0
Factor the quadratic equation:
(x−3)(x+2)=0(x - 3)(x + 2) = 0
This gives two possible solutions:
x=3orx=−2x = 3 \quad \text{or} \quad x = -2
Since x\displaystyle x must be positive (x>0\displaystyle x > 0), we reject x=−2\displaystyle x = -2. Thus:
x=3x = 3
This matches **Option A**.
Note: The textbook answer key incorrectly specifies **Option B** (2\displaystyle 2) as correct. However, our quadratic derivation proves that the only positive root is 3\displaystyle 3 (Option A).
Hence, **Option A** is the correct answer.

About This Chapter: Ratio, Proportion, Indices, Logarithm

Paper

Paper 3: Quantitative Aptitude

Weightage

5-7 Marks

Key Topics

Ratio, Proportion, Indices, Logarithms

This chapter covers Ratio, Proportion, Indices, Logarithms and is part of Paper 3: Quantitative Aptitude in the CA Foundation exam.

View Official ICAI Syllabus

Exam Strategy Tip

This topic carries 5-7 Marks weightage. Focus on understanding core concepts rather than memorizing.

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