Question

The value of \(\sqrt{6+\sqrt{6+\sqrt{6+\sqrt{6+\ldots \infty}}}}\) is : (a) 2 (b) 3 (c) 4 (d) 5

Short answer

a) -2 b) 3 c) 6 d) None of the above Answer: b) 3

Step-by-step solution

Write the recursive formula

Write down the recursive formula for the given nested radical: \(x_n = \sqrt{6 + x_{n-1}}\).

Find the limit of the recursive sequence

To find the value of the infinite nested radical, we need to find the limit of the sequence \(x_n\) as \(n\) approaches infinity. That is, we need to find an \(x\) such that: \(x = \lim_{n \to \infty} x_n = \sqrt{6 + x}\).

Solve the equation

Solve the equation \(x = \sqrt{6 + x}\) for \(x\). To do this, square both sides: \(x^2 = 6 + x\).

Rearrange the equation

Rearrange the equation to the form of a quadratic equation: \(x^2 - x - 6 = 0\).

Factorize or use the quadratic formula

Factorize or use the quadratic formula to find the roots of the equation. By factoring, we get \((x-3)(x+2) = 0\). So, the roots are \(x = 3\) and \(x = -2\).

Choose the correct root

Since we are dealing with a square root, a negative root doesn't make sense in this context. Therefore, the value of the infinite nested radical is the positive root, which is \(x = 3\). The correct answer is (b) 3.

Key concepts

Infinite Nested Radical

An infinite nested radical is a mathematical expression that contains an infinite number of nested square roots. The expression in the exercise, \( \sqrt{6+\sqrt{6+\sqrt{6+\ldots}}} \) is an excellent example of this. To solve such an expression, it's often useful to consider it as a limit and to express it recursively. A recursive formula, like \( x_n = \sqrt{6 + x_{n-1}} \) in the solution, helps define each term of the sequence using the previous term. The challenge with infinite nested radicals is determining if they converge to a finite value and, if so, finding that value.

Understanding and solving infinite nested radicals can enhance problem-solving skills, particularly in the fields of algebra and calculus. By recognizing the pattern and establishing a recursive formula, students can tackle seemingly complex expressions by applying the concepts of recursive sequences and limits.

Quadratic Equation

A quadratic equation is a second-degree polynomial equation of the form \( ax^2 + bx + c = 0 \), where \( a \), \( b \), and \( c \) are constants, and \( a \) is not equal to zero. The quadratic equation appears naturally when solving many algebraic problems, including the infinite nested radical presented in the exercise. Solving a quadratic equation is a fundamental skill in algebra, where methods include factoring, completing the square, or using the quadratic formula. In the given solution, we transformed the recursive equation into a quadratic equation and factored it to find the possible roots. Students need to be familiar with these techniques to determine the correct root that makes sense within the context of the problem.

Recursive Sequences

Recursive sequences are number sequences in which subsequent terms are derived from preceding ones using a fixed rule or formula. In our exercise, the rule is \( x_n = \sqrt{6 + x_{n-1}} \). Recursive sequences can model real-world phenomena such as population growth or the interest compounding in a bank account. By understanding the principle behind recursive sequences, students can predict future values in a sequence and solve complex problems with a systematic approach. Mastery of recursive sequences also provides a strong foundation for studying more advanced mathematical concepts, such as series and mathematical induction.

Limit of a Sequence

The limit of a sequence is the value that the terms of a sequence 'approach' as the number of terms goes to infinity. In many mathematical contexts, finding the limit is crucial for understanding the behavior of sequences over the long term. When dealing with an infinite nested radical—or any recursive sequence—finding the limit can reveal the value that the sequence converges to, if it converges at all. In the context of the exercise, we symbolically expressed this limit as \( x = \lim_{n \to \infty} x_n \), which captures the essence of the sequence's behavior as \( n \) becomes infinitely large. Through the exercise, students practice not only finding limits but also recognizing convergence and divergence in sequences, which is pivotal for calculus and higher-level mathematics.