Problem 14 - Algebra for JEE ADVANCED

Question

Suppose that \(F(n+1)=\frac{2 F(n)+1}{2}\) for \(n=1,2,3, \ldots\) and \(F(1)=2\). Then, \(F(101)\) equals a. 50 b. 52 c. 54 d. none of these

Step-by-Step Solution

Answer

\(F(101) = 52\). Answer is b.

Step 1: Understand the Recursive Formula

The problem states a recursive formula: \(F(n+1) = \frac{2F(n)+1}{2} \) with an initial condition \(F(1) = 2\). This means that each term in the sequence is defined in terms of the previous term.

Step 2: Calculate the First Few Terms

Start by calculating the next few terms in the sequence using the formula.- For \(F(2)\), use \(F(n+1)=\frac{2F(1)+1}{2}\): \[F(2) = \frac{2 \times 2 + 1}{2} = \frac{5}{2}\]- For \(F(3)\), use \(F(n+1)=\frac{2F(2)+1}{2}\): \[F(3) = \frac{2 \times \frac{5}{2} + 1}{2} = \frac{6}{2} = 3\]- Calculate one more, \(F(4)\) to observe any pattern: \[F(4) = \frac{2 \times 3 + 1}{2} = \frac{7}{2}\]

Step 3: Observe and Extend the Pattern

Observe that the sequence seems to be forming a pattern involving fractions whose numerators increase by 1 every two steps.Notice:- \(F(1) = 2 = \frac{4}{2}\)- \(F(2) = \frac{5}{2}\)- \(F(3) = \frac{6}{2} = 3\)- \(F(4) = \frac{7}{2}\)The numerator appears to increment by 1. Let's define the pattern for the numerator as \(n+3\).

Step 4: Generalize the Expression

Since each step increases the numerator by 1 while the denominator remains 2 when simplified, a general expression for \(F(n)\) can be defined:- \(F(n) = \frac{n+3}{2}\).Check it:- \(F(1) = \frac{1+3}{2} = \,2\)- \(F(2) = \frac{2+3}{2} = \, \frac{5}{2}\)- \(F(3) = \frac{3+3}{2} = \,3\)

Step 5: Calculate \(F(101)\)

Apply the generalized formula to find \(F(101)\):\[F(101) = \frac{101 + 3}{2} = \frac{104}{2} = 52\].

Key Concepts

Sequences

Sequences are an ordered list of numbers that follow a specific rule or pattern. They can be finite or infinite, depending on whether they continue indefinitely or have an endpoint.
In our original exercise, the numbers generated by the recursive formula form a sequence. This sequence provides a way to generate numbers in a specific order, starting from the initial condition, which is given as the first term of the sequence.
Understanding sequences is crucial because they help identify consistent mathematical patterns, allowing us to predict and calculate future numbers in the sequence.
Common types of sequences include:

• Arithmetic sequences: where each term is obtained by adding a constant to the previous term.
• Geometric sequences: where each term is obtained by multiplying the previous term by a constant.
• Recursive sequences: where each term is defined based on one or more preceding terms, as seen in our original exercise.

Recognizing these patterns is key to determining the right sequence type and solving problems effectively.

Recursive Formula

A recursive formula is a method of defining sequences, allowing each term to be expressed in terms of its predecessors. This is fundamental in scenarios where mathematical expressions evolve based on prior values.
In the problem presented, the recursive formula is given as: \( F(n+1) = \frac{2F(n)+1}{2} \). This indicates that each subsequent term is calculated using the value of the previous term. With the initial value provided, \( F(1) = 2 \), we can proceed to find successive terms.
Recursive formulas are advantageous because:

• They simplify the process of finding terms in a sequence without needing a direct formula for terms a long way into the sequence.
• They often reveal mathematical relationships or patterns that are not immediately obvious.

However, it's important to meticulously apply the recursive steps as any small error can propagate through subsequent calculations.
Understanding how to work with recursive formulas allows us to unlock complex sequences and appreciate the mathematical interconnections they represent.

Mathematical Patterns

Mathematical patterns are consistent or recurring ways that numbers behave within equations, sequences, or mathematical systems. Spotting and understanding these patterns is essential in solving complex problems and predicting future outcomes.
In the context of the original exercise, after calculating several terms, a clear pattern emerged in the numerators, incrementing by 1 each step. This insight allowed for the creation of a general formula for \( F(n) = \frac{n+3}{2} \).
Mathematical patterns help simplify problems by:

• Providing an intuitive grasp of the problem structure and enabling predictions about unknown values, as evidenced in the simplified expression for \( F(101) \).
• Facilitating problem-solving by helping develop general solutions from specific instances, making complex problems more manageable.

By practicing pattern identification, students can enhance their mathematical intuition, transforming seemingly disparate problems into familiar concepts with recognized solutions. This skill is invaluable across various fields of mathematics and science.