Question

If \(t_{1}=2, t_{2}=2\) and \(t_{n-1}=t_{n}+1\) for \(n \geq 3\) then \(t_{5}=\) (a) 1 (b) 0 (c) \(-1\) (d) 5

Short answer

Answer: The 5th term of the sequence is -1.

Step-by-step solution

Identify the given sequence properties

The given sequence has two initial conditions: \(t_1=2\) \(t_2=2\) And a recurrence relation for \(n\ge 3\): \(t_{n-1}=t_{n}+1\)

Find t_3

We can use the recurrence relation to find the third term by replacing n = 3: \(t_{3-1} = t_3 + 1\) \(t_2 = t_3 + 1\) Now, using the initial condition \(t_2=2\): \(2 = t_3 + 1\) Solving for \(t_3\): \(t_3 = 2 - 1 = 1\)

Find t_4

Similarly, to find the fourth term, we replace n = 4 in the given recurrence relation: \(t_{4-1} = t_4 + 1\) \(t_3 = t_4 + 1\) Using the value of \(t_3\) that we found in Step 2: \(1 = t_4 + 1\) Solving for \(t_4\): \(t_4 = 1 - 1 = 0\)

Find t_5

Finally, we find the fifth term, \(t_5\), by replacing n = 5 in the recurrence relation: \(t_{5-1} = t_5 + 1\) \(t_4 = t_5 + 1\) We substitute the value of \(t_4\) found in Step 3: \(0 = t_5 + 1\) Solving for \(t_5\): \(t_5 = 0 - 1 = -1\) Thus, \(t_5 = -1\), which corresponds to option (c).

Key concepts

Understanding Sequence Properties

A sequence is essentially an ordered list of numbers, each of which is typically defined by a specific rule or formula. Sequence properties refer to the characteristics that define and distinguish one sequence from another. In the problem, we are dealing with a linear recurrence relation, indicating a sequence defined by its previous terms.
The provided sequence has notable properties: the sequence starts with two initial terms, and each subsequent term is determined by the initial conditions and the recurrence relation. Understanding these properties allows us to predict any term in the sequence and solve problems involving sequence manipulation efficiently.
In mathematical sequences such as these, the importance of recognizing the specific pattern or rule that applies assumes prominence. Each sequence can either converge, indicating that it approaches a fixed number, or diverge, growing indefinitely or oscillating without settling on any one number. Recognizing these characteristics is vital in deducing the behavior of the entire sequence.

Importance of Initial Conditions

Initial conditions in a sequence lay the foundation for the entire succession of terms. These are the predefined starting points, and in our exercise, they are given as \( t_1 = 2 \) and \( t_2 = 2 \). Knowing these initial terms is critical because they feed directly into the recurrence relation, allowing us to predict future terms.
Initial conditions also help in uniquely determining a sequence from potentially infinite possibilities. This is because, without initial conditions, a recurrence relation might not uniquely specify a sequence.
For students preparing for competitive exams, understanding initial conditions enhances problem-solving capabilities, enabling them to approach problems systematically by identifying what they know at the outset.

Techniques for Solving Recurrence Relations

Recurrence relations are equations that define a sequence of numbers. In this exercise, the relation is \( t_{n-1} = t_n + 1 \). Solving such relations involves finding an explicit formula for the terms or calculating a specific term using known values.
The technique usually entails substituting the known values from the initial conditions into the recurrence relation and solving for the unknown term. For example:

• Find \( t_3 \) by using \( t_2 = t_3 +1 \), resulting in \( t_3 = 1 \).
• Similarly, find \( t_4 \) and \( t_5 \) by iteratively using the relation.

This step-by-step substitution and solving are key techniques used in many mathematical problems, particularly those in exam settings like IIT-JEE Mathematics. They require a systematic approach, patience, and an eye for detail.

Role of Recurrence Relations in IIT-JEE Mathematics

The IIT-JEE Mathematics syllabus often includes questions about sequences for a critical reason: they test a student's ability to apply mathematical logic and problem-solving skills under time constraints.
In IIT-JEE, recurrence relations find relevance not just as a standalone topic but also integrated with other areas, such as calculus or algebra. Solving them can help develop a deeper understanding of mathematical principles. Questions on this topic often require swift pattern recognition, adept substitution skills, and precision.
Preparing for such questions requires practice. Here, the emphasis is often on the application of known relationships - one builds upon the last to form a cohesive and linked series of steps, leading to the correct answer. The ability to break down these steps in an organized manner is crucial for exam success.