EquationsMCQMTP Mar 21Question 1099 of 221
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The value of 6+6+6+......∞\displaystyle \sqrt{6 + \sqrt{6 + \sqrt{6 + ...... \infty}}}

Options

A−3\displaystyle -3
B2\displaystyle 2
C3\displaystyle 3
D4\displaystyle 4
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Correct Answer

✅ Option c — 3\displaystyle 3

All Options:

  • A−3\displaystyle -3
  • B2\displaystyle 2
  • C3\displaystyle 3
  • D4\displaystyle 4

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Detailed Solution & Explanation

Let y=6+6+6+…∞\displaystyle y = \sqrt{6 + \sqrt{6 + \sqrt{6 + \dots \infty}}}.

Since the sequence extends infinitely under the square root, we can write:
y=6+yy = \sqrt{6 + y}

Squaring both sides to eliminate the square root:
y2=6+yy^2 = 6 + y
y2−y−6=0y^2 - y - 6 = 0

Factoring the quadratic equation:
y2−3y+2y−6=0y^2 - 3y + 2y - 6 = 0
y(y−3)+2(y−3)=0y(y - 3) + 2(y - 3) = 0
(y−3)(y+2)=0(y - 3)(y + 2) = 0

This gives two possible solutions:
y=3\displaystyle y = 3 or y=−2\displaystyle y = -2.

Since y\displaystyle y represents a positive square root of positive numbers, y\displaystyle y must be positive (y>0\displaystyle y > 0). Therefore, we discard the negative value y=−2\displaystyle y = -2.
Thus, y=3\displaystyle y = 3.

Hence, the correct option is **Option (c)**.

About This Chapter: Equations

Paper

Paper 3: Quantitative Aptitude

Weightage

4-6 Marks

Key Topics

Linear, Quadratic and Cubic Equations

This chapter covers Linear, Quadratic and Cubic Equations and is part of Paper 3: Quantitative Aptitude in the CA Foundation exam.

View Official ICAI Syllabus

Exam Strategy Tip

This topic carries 4-6 Marks weightage. Focus on understanding core concepts rather than memorizing.

More Questions from Equations

If 3x+y+2x−y=−1\displaystyle \frac{3}{x+y} + \frac{2}{x-y} = -1 and 1x+y−1x−y=43\displaystyle \frac{1}{x+y} - \frac{1}{x-y} = \frac{4}{3} then (x,y)\displaystyle (x, y) is:

2x+510+3x+1015=5\displaystyle \frac{2x+5}{10} + \frac{3x+10}{15} = 5, find x\displaystyle x

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The cost of 2\displaystyle 2 oranges and 3\displaystyle 3 apples is 28\displaystyle 28. If the cost of an apple is doubled then the cost of 3\displaystyle 3 oranges and 5\displaystyle 5 apples is 75\displaystyle 75. The original cost of 7\displaystyle 7 oranges and 4\displaystyle 4 apples (in Rs) is:

In a multiple choice question paper consisting of 100\displaystyle 100 questions of 1\displaystyle 1 mark each, a candidate gets 60%\displaystyle 60\% marks. If the candidate attempted all questions and there was a penalty of 0.25\displaystyle 0.25 marks for wrong answers is:

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