Problem 17 - A Complete Resource Book in Mathematics for JEE Main

Question

If \(x_{n+1}=\sqrt{\frac{1+x_{n}}{2}}\), then \(\cos \left(\frac{\sqrt{1-x_{0}^{2}}}{x_{1} x_{2} x_{3} \ldots . \text { to } \infty}\right)\left(-1

Step-by-Step Solution

Answer

The answer is (A) \(x_0\).

Step 1: Understand the iterative formula

The given formula is \(x_{n+1} = \sqrt{\frac{1+x_{n}}{2}}\). It's a sequence starting with \(x_0\) where \(-1

Step 2: Explore the sequence behavior

Notice that if \(x_n = \cos(\theta)\), then \(x_{n+1} = \cos(\theta/2)\) due to the identity \(\cos(\theta/2) = \sqrt{ \frac{1 + \cos(\theta)}{2} }\). Thus, the sequence reduces the angle by half each iteration.

Step 3: Evaluate the infinite product

The infinite product \(x_1 x_2 x_3 \ldots\) can be transformed into \(\cos(\theta/2) \cos(\theta/4) \ldots\). This infinite product equals \(\cos(\theta/2^n)\) as \(n \to \infty\), leading it to approach \(1\) because \(\cos(0) = 1\).

Step 4: Substitute and simplify the expression

The expression given is \(\cos\left(\frac{\sqrt{1-x_{0}^{2}}}{\prod_{n=1}^{\infty} x_n}\right)\). With our result from Step 3, the denominator simplifies to \(1\), so the expression inside the cosine function is \(\sqrt{1-x_0^2}\) or \(\sin(\theta)\).

Step 5: Determine the cosine value

We know that \(\sqrt{1-x_0^2} = \sin(\theta)\), hence \(\cos(\sin(\theta)) = x_0\). So the expression evaluated simplifies to \(x_0\).

Key Concepts

Sequence Convergence

When dealing with sequences, especially iterative ones defined by formulas like \( x_{n+1} = \sqrt{\frac{1+x_{n}}{2}} \), an important aspect to analyze is how the sequence behaves as the number of iterations goes to infinity. In this case, the sequence starts with an initial value \( x_0 \) where \( -1 < x_0 < 1 \). By plugging each term back into the formula, we generate a series of values \( x_1, x_2, x_3, \ldots \).
The process of determining what value, if any, this sequence settles on over time is called 'sequence convergence'. This convergence analysis can often reveal a specific value the sequence approaches, known as the limit.
For this sequence, we observe the behavior through trigonometric identities. When \( x_n = \cos(\theta) \), it demonstrates a unique pattern. Through the formula \( x_{n+1} = \cos\left(\frac{\theta}{2}\right) \), each iteration halves the angle, making the sequence get closer and closer to zero in terms of angle, or simply approaching the value of the cosine function as the angle approaches zero.

Cosine Function

The cosine function plays a fundamental role in trigonometry and is paramount in understanding the iterative sequence given by \( x_{n+1} = \sqrt{\frac{1+x_{n}}{2}} \). The formula is a derivation of the identity \( \cos\left(\frac{\theta}{2}\right) = \sqrt{\frac{1 + \cos(\theta)}{2}} \). This identity is used to describe how the cosine of half an angle relates to the cosine of the full angle.
The cosine function oscillates between -1 and 1 and is periodic with a period of \( 2\pi \). This makes it ideal to analyze repetitive or cyclic patterns in trigonometric sequences.
In the context of the sequence, when \( x_n = \cos(\theta) \), we see that with each iteration the angle \( \theta \) is reduced by half, which keeps the values within the cosine's range of -1 to 1. The beauty of this transformation is that it can infinitely approach a fixed point, namely \( \cos(0) = 1 \), which is noted as the sequence progresses to infinity. This behavior helps us solve the given problem by transforming the sequence into an infinite product linked to trigonometric identities.

Iterative Formula

An iterative formula, like \( x_{n+1} = \sqrt{\frac{1+x_{n}}{2}} \), provides a recursive way to generate a sequence of numbers. This type of formula depends on the previous term to define the next one. Such formulas are widely used in mathematical problems to explore behaviors of sequences and investigate properties like convergence or divergence.
In this particular problem, understanding how each term transforms is crucial. With \( x_{n+1} \) being derived from \( x_n \) as \( \cos(\theta/2) \), it allows us to see more clearly how each iteration affects the angle \( \theta \). As it progressively halves the angle, this iterative process reveals insights into the sequence's long-term behavior.
Key characteristics of iterative formulas include:

• Dependency on prior values to define future terms.
• A focus on discovering the limiting behavior as the sequence approaches infinity.
• Connections to known mathematical identities, like the cosine function, to simplify complex evaluations.

Understanding iterative formulas is essential for tackling complex sequences and unlocking the secrets of their convergence or divergence across various mathematical problems.

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Problem 16

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Problem 18

Other exercises in this chapter

Problem 15

The inequality \(2^{\sin \theta}+2^{\cos \theta} \geq 2^{\left(t-\frac{1}{\sqrt{2}}\right)}\) holds for (A) \(0 \leq \theta

View solution

Problem 16

The expression \(2^{\sin \theta}+2^{-\cos \theta}\) is minimum when \(\theta\) is equal to (A) \(2 n \pi+\frac{\pi}{4}, n \in I\) (B) \(2 n \pi+\frac{7 \pi}{4},

View solution

Problem 18

If \(0

View solution

Problem 19

If \(\cos (\theta-\alpha)=a\) and \(\sin (\theta-\beta)=b(0

View solution

Key Concepts

Sequence Convergence

When dealing with sequences, especially iterative ones defined by formulas like \( x_{n+1} = \sqrt{\frac{1+x_{n}}{2}} \), an important aspect to analyze is how the sequence behaves as the number of iterations goes to infinity. In this case, the sequence starts with an initial value \( x_0 \) where \( -1 < x_0 < 1 \). By plugging each term back into the formula, we generate a series of values \( x_1, x_2, x_3, \ldots \).
The process of determining what value, if any, this sequence settles on over time is called 'sequence convergence'. This convergence analysis can often reveal a specific value the sequence approaches, known as the limit.
For this sequence, we observe the behavior through trigonometric identities. When \( x_n = \cos(\theta) \), it demonstrates a unique pattern. Through the formula \( x_{n+1} = \cos\left(\frac{\theta}{2}\right) \), each iteration halves the angle, making the sequence get closer and closer to zero in terms of angle, or simply approaching the value of the cosine function as the angle approaches zero.

Cosine Function

The cosine function plays a fundamental role in trigonometry and is paramount in understanding the iterative sequence given by \( x_{n+1} = \sqrt{\frac{1+x_{n}}{2}} \). The formula is a derivation of the identity \( \cos\left(\frac{\theta}{2}\right) = \sqrt{\frac{1 + \cos(\theta)}{2}} \). This identity is used to describe how the cosine of half an angle relates to the cosine of the full angle.
The cosine function oscillates between -1 and 1 and is periodic with a period of \( 2\pi \). This makes it ideal to analyze repetitive or cyclic patterns in trigonometric sequences.
In the context of the sequence, when \( x_n = \cos(\theta) \), we see that with each iteration the angle \( \theta \) is reduced by half, which keeps the values within the cosine's range of -1 to 1. The beauty of this transformation is that it can infinitely approach a fixed point, namely \( \cos(0) = 1 \), which is noted as the sequence progresses to infinity. This behavior helps us solve the given problem by transforming the sequence into an infinite product linked to trigonometric identities.

Iterative Formula

An iterative formula, like \( x_{n+1} = \sqrt{\frac{1+x_{n}}{2}} \), provides a recursive way to generate a sequence of numbers. This type of formula depends on the previous term to define the next one. Such formulas are widely used in mathematical problems to explore behaviors of sequences and investigate properties like convergence or divergence.
In this particular problem, understanding how each term transforms is crucial. With \( x_{n+1} \) being derived from \( x_n \) as \( \cos(\theta/2) \), it allows us to see more clearly how each iteration affects the angle \( \theta \). As it progressively halves the angle, this iterative process reveals insights into the sequence's long-term behavior.
Key characteristics of iterative formulas include:

• Dependency on prior values to define future terms.
• A focus on discovering the limiting behavior as the sequence approaches infinity.
• Connections to known mathematical identities, like the cosine function, to simplify complex evaluations.

Understanding iterative formulas is essential for tackling complex sequences and unlocking the secrets of their convergence or divergence across various mathematical problems.