EquationsPYQ Jan 26Question 4502 of 155
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The value of is 5+5+5+…∞\displaystyle \sqrt{5+\sqrt{5+\sqrt{5+\dots\infty}}} is

Options

A0
B1+212\displaystyle \frac{1+\sqrt{21}}{2}
C1
D−1+212\displaystyle \frac{-1+\sqrt{21}}{2}
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Correct Answer

✅ Option b — 1+212\displaystyle \frac{1+\sqrt{21}}{2}

All Options:

  • A0
  • B1+212\displaystyle \frac{1+\sqrt{21}}{2}
  • C1
  • D−1+212\displaystyle \frac{-1+\sqrt{21}}{2}

Detailed Solution & Explanation

Let y\displaystyle y be the value of the given expression:
y=5+5+5+…∞y = \sqrt{5+\sqrt{5+\sqrt{5+\dots\infty}}}

Since the expression is infinite, we can substitute y\displaystyle y for the nested radical term:
y=5+yy = \sqrt{5+y}

Squaring both sides of the equation (noting that y>0\displaystyle y > 0 since it is a positive square root):
y2=5+yy^2 = 5 + y
y2−y−5=0y^2 - y - 5 = 0

This is a quadratic equation in the form ay2+by+c=0\displaystyle ay^2 + by + c = 0, where a=1\displaystyle a = 1, b=−1\displaystyle b = -1, and c=−5\displaystyle c = -5. Using the quadratic formula:
y=−b±b2−4ac2ay = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
y=−(−1)±(−1)2−4(1)(−5)2(1)y = \frac{-(-1) \pm \sqrt{(-1)^2 - 4(1)(-5)}}{2(1)}
y=1±1+202y = \frac{1 \pm \sqrt{1 + 20}}{2}
y=1±212y = \frac{1 \pm \sqrt{21}}{2}

Since y\displaystyle y must be positive, we discard the negative root 1−212\displaystyle \frac{1 - \sqrt{21}}{2} (as 21≈4.58\displaystyle \sqrt{21} \approx 4.58, making 1−21<0\displaystyle 1-\sqrt{21} < 0). Thus:
y=1+212y = \frac{1+\sqrt{21}}{2}

Hence, **Option B** is the correct answer.

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

Find value of x2−10x+1\displaystyle x^2 - 10x + 1 if x=15−26\displaystyle x = \frac{1}{5-2\sqrt{6}}

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:

The values of x\displaystyle x and y\displaystyle y satisfying the equations 3x+y+2x−y=3\displaystyle \frac{3}{x+y} + \frac{2}{x-y} = 3 and 2x+y+3x−y=233\displaystyle \frac{2}{x+y} + \frac{3}{x-y} = \frac{23}{3} given by

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