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Introduction

What is a sequence?

Informal Definition

A sequence is an ordered list of mathematical objects, such as real numbers, complex numbers, matrices, or any other mathematical objects, arranged in a specific order separated by commas. Sequences can be finite, with a limited number of terms, or infinite, extending indefinitely.

The order of the elements in a sequence is crucial, and each element is called a term of the sequence. For example, the sequence of powers of 2, starting from \(1\), is an infinite sequence \(1, 2, 4, 8, 16, \ldots\), where each term after the first is obtained by multiplying the previous term by \(2\).

Formal Definition

Formally, a sequence can be defined as a function \(f: D \rightarrow A\), where \(D\) is a domain consisting of either the first \(n\) positive integers \(\{1, 2, 3, \ldots, n\}\) for a finite sequence of length \(n\), or the set of all positive integers \(\mathbb{N} = \{1, 2, 3, \ldots\}\) for an infinite sequence. The codomain \(A\) can be any set, such as \(\mathbb{R}\) (the set of real numbers), \(\mathbb{C}\) (the set of complex numbers), or the set of all matrices of a given size or any other set of objects. Element \(f(n)\) in the codomain is the nth term of the sequence.

Example

The sequence \(1, 2, 4, 8, 16, \ldots\) is an infinite sequence generated by the function \(f(n) = 2^n\) for \(n \in \mathbb{N}\). Here, \(D = \mathbb{N}\) and \(A = \mathbb{R}\). For each \(n \in D\), \(f(n) = 2^n\) determines the value of the nth term in the sequence. This sequence showcases how each term is related to its position \(n\) in the sequence through a specific mathematical rule, in this case, exponentiation.

Notation

Sequences are often represented using specific notation to succinctly describe their structure and elements. A common notation for sequences is \((a_n)\), where \(a_n\) represents the nth term of the sequence.

For finite sequences, we might specify the range of \(n\) explicitly, such as \((a_n)_{n=1}^{10}\), indicating the sequence starts at \(n=1\) and ends at \(n=10\). For example, the sequence of the first ten positive even numbers could be denoted as \((2n)_{n=1}^{10}\), explicitly listing the terms as \(2, 4, 6, 8, 10, 12, 14, 16, 18, 20\). Similarly, an infinite sequence might be denoted simply as \((a_n)_{n=1}^{\infty}\), such as \((2^n)_{n=1}^{\infty}\) for the sequence \(2, 4, 8, 16, \ldots\), indicating it continues indefinitely following the given pattern. This notation is powerful for both its precision in defining the terms of the sequence and its flexibility in representing a wide range of sequences, from simple arithmetic progressions to complex formula-based sequences.

  1. \((n^2)_{n=1}^{\infty}\) represents the sequence of square numbers: \(1, 4, 9, 16, 25, \ldots\), where each term is the square of its position in the sequence, continuing indefinitely.

  2. \(\left(\frac{1}{n}\right)_{n=1}^{\infty}\) denotes the sequence of reciprocals of natural numbers: \(1, \frac{1}{2}, \frac{1}{3}, \frac{1}{4}, \ldots\), illustrating a sequence that approaches zero as \(n\) increases without bound.

  3. \(((-1)^n)_{n=1}^{\infty}\) describes the alternating sequence: \(-1, 1, -1, 1, -1, \ldots\), where each term switches sign from the previous term, creating a pattern of negative and positive ones.

  4. \((3n + 1)_{n=1}^{\infty}\) gives the sequence: \(4, 7, 10, 13, 16, \ldots\), which is an arithmetic sequence with a common difference of \(3\), starting from \(4\).

  5. \((2^{n} - n)_{n=1}^{\infty}\) generates the sequence: \(1, 2, 5, 12, 27, \ldots\), where each term is the result of subtracting the position \(n\) from \(2\) raised to the power of \(n\), showcasing a rapidly increasing sequence as \(n\) grows.

  6. \((n^2)_{n=1}^{5}\) represents the sequence of square numbers for the first five natural numbers: \(1, 4, 9, 16, 25\). This sequence contains the squares of the numbers \(1\) through \(5\), ending after the fifth term.

  7. \(\left(\frac{1}{n}\right)_{n=1}^{4}\) denotes the sequence of reciprocals of the first four natural numbers: \(1, \frac{1}{2}, \frac{1}{3}, \frac{1}{4}\). This sequence showcases the reciprocals starting from \(1\) to \(\frac{1}{4}\), and it concludes at the fourth term.

  8. \((2n - 1)_{n=1}^{6}\) gives the sequence of the first six odd numbers: \(1, 3, 5, 7, 9, 11\). This arithmetic sequence is generated by multiplying \(n\) by \(2\) and subtracting \(1\), demonstrating a consistent pattern of odd numbers up to the sixth term.

Index of a sequence

The index of a sequence refers to the position of an element within the sequence. It acts as an identifier for each term in the sequence, indicating its order or placement. For sequences defined mathematically, the index is typically denoted by a natural number (such as \(n\), \(i\), or \(k\)) in the set \(\mathbb{N}\) (which can include \(0\) if the sequence starts from \(0\)), and it serves as the argument to the sequence's defining function.

For example, in the sequence \((a_n)_{n=1}^{\infty}\), \(n\) is the index of the sequence. It starts from \(1\) and increases by \(1\) for each subsequent term, indicating the progression of the sequence. The term \(a_n\) represents the element of the sequence at position \(n\). The index allows for the direct reference to any term in the sequence, enabling the calculation or identification of sequence elements based on their position.

Methods to define a sequence

Sequences can be defined in several ways, with the most common methods being through an explicit formula or a recursive definition. Each method provides a systematic approach to generating the terms of a sequence, catering to different kinds of patterns and complexities.

Explicit Formula

An explicit formula defines the nth term of a sequence directly in terms of \(n\), without relying on the knowledge of previous terms. It provides a direct rule to calculate any term in the sequence given its position.

Example: Consider the sequence \(1, 3, 9, 27, \ldots\), where each term is three times the previous term, starting from \(1\). The explicit formula for this sequence is \(a_n = 3^{n-1}\), where \(n\) is the position of the term in the sequence. This formula allows for the direct computation of any term in the sequence without needing to calculate the preceding terms. For instance, to find the 5th term, we simply compute \(a_5 = 3^{5-1} = 81\).

Recursive Definition

A recursive definition specifies each term of a sequence using one or more previous terms, starting with one or more initial terms. This approach defines a sequence by its relationship between successive terms.

Example: Using the same sequence as above \(1, 3, 9, 27, \ldots\), we can define it recursively by stating that each term is three times the previous term, with the first term given as \(a_1 = 1\). The recursive formula is then \(a_n = 3a_{n-1}\) for \(n > 1\). This means to find a specific term, you start from the initial term and use the formula to find subsequent terms iteratively. For example, to find the 4th term, you would start with \(a_1 = 1\), then find \(a_2 = 3 \cdot 1 = 3\), then \(a_3 = 3 \cdot 3 = 9\), and finally \(a_4 = 3 \cdot 9 = 27\).

Both methods have their advantages and applications. The explicit formula is useful for calculating terms without iterating through the sequence, ideal for finding terms far into the sequence. The recursive definition is beneficial when the relationship between consecutive terms is simple to describe or when the sequence is naturally defined by its progression from one term to the next.

Fibonacci Sequence

Fibonacci Sequence: Recursive Definition

The Fibonacci sequence is defined recursively as follows:

  • Base cases: \(a_1 = 0, a_2 = 1\)
  • Recursive step: For \(n > 2\), \(a_n = a_{n-1} + a_{n-2}\)

This means each term is the sum of the two preceding terms, starting with \(0\) and \(1\).

Let us find the first 10 terms of the Fibonacci sequence. We apply the recursive definition:

  1. \(a_1 = 0\)
  2. \(a_2 = 1\)
  3. \(a_3 = a_2 + a_1 = 1 + 0 = 1\)
  4. \(a_4 = a_3 + a_2 = 1 + 1 = 2\)
  5. \(a_5 = a_4 + a_3 = 2 + 1 = 3\)
  6. \(a_6 = a_5 + a_4 = 3 + 2 = 5\)
  7. \(a_7 = a_6 + a_5 = 5 + 3 = 8\)
  8. \(a_8 = a_7 + a_6 = 8 + 5 = 13\)
  9. \(a_9 = a_8 + a_7 = 13 + 8 = 21\)
  10. \(a_{10} = a_9 + a_8 = 21 + 13 = 34\)

So, the first 10 terms of the Fibonacci sequence are: \(0, 1, 1, 2, 3, 5, 8, 13, 21, 34\).

Binet's Formula

Binet's formula offers a non-recursive way to calculate the nth Fibonacci number directly:

\[a_n = \frac{\phi^n - (1-\phi)^n}{\sqrt{5}}\]

where \(\phi = \frac{1 + \sqrt{5}}{2}\) is the golden ratio. Unlike the recursive definition, which requires calculating all previous terms, Binet's formula allows for the direct computation of any term in the sequence. This method is particularly useful for finding terms far into the sequence without iterating through all preceding numbers. We will learn how to derive this formula later on.

Convergence and Divergence

A converging sequence is one that settles into a particular pattern as you go further along its terms, eventually getting closer and closer to a specific value, called the limit. The idea is that as you progress through the sequence, the difference between each term and this limit becomes smaller and smaller, effectively becoming negligible. An example of a converging sequence is \(\frac{1}{n}\), which approaches \(0\) as \(n\) becomes very large. Another example: consider the sequence \(\left(\frac{1}{2^n}\right)_{n=1}^{\infty}\): \( \frac{1}{2}, \frac{1}{4}, \frac{1}{8}, \frac{1}{16}, \ldots \)

This sequence converges to \(0\) because as \(n\) increases, \(\frac{1}{2^n}\) gets smaller and smaller, approaching \(0\) more closely with each successive term.

A diverging sequence, on the other hand, doesn't settle down to a single value as you move along its terms. It might keep growing larger and larger, oscillate between values, or behave unpredictably without nearing any specific number. For instance, the sequence \(n\), which simply counts upwards (1, 2, 3, ...), is divergent because it gets indefinitely large as you go further along the sequence.

Look at the sequence \(((-1)^n)_{n=1}^{\infty}\): \( -1, 1, -1, 1, \ldots \)

This sequence diverges because it does not approach a single value; instead, it oscillates between \(-1\) and \(1\) indefinitely, never settling down to a specific limit.

We will not go into more discussion on convergence and divergence as this topic has NOT been included in JEE Mains and Advanced.

Series

A series is the sum of the terms of a sequence. It is often represented as the summation of sequence terms \(a_n\) over a specified range of values. If the sequence is infinite, the series is called an infinite series and is written as:

\[S = a_1 + a_2 + a_3 + \ldots \]

For a finite sequence, the series sums over a finite number of terms, and can be expressed as:

\[S_n = a_1 + a_2 + \ldots + a_n \]

where \(n\) is the index of the last term in the series. For examples:

  1. For the sequence \((\frac{1}{j})_{j=1}^n\), summing these terms, we denote the sum as

    \[S_n = \frac{1}{1} + \frac{1}{2} + \ldots + \frac{1}{n}\]

  2. For the sequence \((\frac{k}{k+1})_{k=0}^{n-1}\), the sum is represented as

    \[S_n = \frac{0}{1} + \frac{1}{2} + \ldots + \frac{n-1}{n}\]

  3. For the sequence \((r^2 + r + 1)_{r=0}^{n-1}\), the sum is expressed as

    \[S_n = (0^2 + 0 + 1) + (1^2 + 1 + 1) + \ldots + ((n-1)^2 + (n-1) + 1)\]
    \[\implies S_n = 1 + 3 + 7 + 13 + \ldots + \left((n-1)^2 + (n-1) + 1\right)\]

A bit more formal definition of series goes like this:

Definition (Series and Partial Sums):

Let \( (t_n) \) denote a sequence of real numbers. A series generated by \( (t_n) \) is the ordered collection of sums known as partial sums, and is denoted by \( S = (s_k) \), where each \( s_k \) is defined as follows:

\[ s_1 := t_1 \]
\[ s_2 := s_1 + t_2 = (t_1 + t_2) \]
\[ \vdots \]
\[ s_k := s_{k-1} + t_k = (t_1 + t_2 + \cdots + t_k) \]

Explanation of Partial Sums:

  1. Initial Partial Sum (\( s_1 \)): The series commences with the first partial sum \( s_1 \), which is equal to the first term of the sequence, \( t_1 \).

  2. Subsequent Partial Sums (\( s_2, s_3, \ldots \)): Each subsequent partial sum is obtained by adding the next term in the sequence to the previous partial sum. Thus, the second partial sum \( s_2 \) is the sum of the first two terms, the third partial sum \( s_3 \) is the sum of the first three terms, and so forth.

  3. General Partial Sum (\( s_k \)): The k-th partial sum \( s_k \) represents the sum of the first \( k \) terms of the sequence. It encapsulates the cumulative addition of sequence terms up to the k-th position.

Convergence and Divergence of Series:

  • Convergence: If the sequence of partial sums \( (s_k) \) approaches a finite limit as \( k \) increases without bound, the series is said to converge, and the limit is known as the sum of the series.

  • Divergence: If the sequence of partial sums does not approach a finite limit, the series is described as divergent.

Again we will not dive into convergence and divergence more than this, but still it is important for our understanding of series. You can get more idea from the following examples.

Example

Let's consider the sequence \( (t_n) \) where each term is defined by \( t_n = \frac{1}{n} \). This is often referred to as the harmonic sequence.

Using the sequence \( (t_n) \), we can form a series, which is the sum of the terms of the sequence. The series generated by the harmonic sequence is:

\[ s_1 = \frac{1}{1} \]
\[ s_2 = \frac{1}{1} + \frac{1}{2} \]
\[ s_3 = \frac{1}{1} + \frac{1}{2} + \frac{1}{3} \]
\[ \vdots \]
\[ s_k = \frac{1}{1} + \frac{1}{2} + \frac{1}{3} + \ldots + \frac{1}{k} \]

The series formed by this sequence is the harmonic series. The \( k \)-th partial sum \( s_k \) is the sum of the reciprocals of the first \( k \) positive integers.

It is a well-established result that the harmonic series diverges, which means that the sequence of its partial sums \( (s_k) \) does not converge to a finite limit as \( k \) tends to infinity. The terms \( \frac{1}{n} \) get smaller as \( n \) gets larger, but they do not diminish quickly enough for the sum to stabilize at a finite number. Thus, despite the terms approaching zero, the sum grows without bound.

Consider the sequence \( (t_n) \) where each term is given by \( t_n = \frac{1}{2^n} \). This sequence represents the terms of a geometric series with a common ratio of \( \frac{1}{2} \).

The series generated by this sequence is:

\[ s_1 = \frac{1}{2} \]
\[ s_2 = \frac{1}{2} + \frac{1}{4} \]
\[ s_3 = \frac{1}{2} + \frac{1}{4} + \frac{1}{8} \]
\[ \vdots \]
\[ s_k = \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \ldots + \frac{1}{2^k} \]

This series converges to 1. We can prove this geometrically:

Imagine a square with an area of 1. If you cut this square in half, one of the halves represents an area of \( \frac{1}{2} \), which corresponds to the first term of our series, \( s_1 \).

Now, if you take the remaining half and cut it in half again, you get a quarter of the original square, which represents an area of \( \frac{1}{4} \), corresponding to the second term of our series, \( s_2 \).

Continuing this process, each subsequent cut divides the remaining area in half again, corresponding to the terms \( \frac{1}{8}, \frac{1}{16}, \) and so on. The total area never exceeds the area of the original square, so the series converges to 1.

This geometric progression can be visualized by continuing to halve a rectangle or square, fitting each subsequent rectangle within the remaining space, and never exceeding the total area. It's a simple yet powerful illustration of an infinite process approaching a finite limit.

Sequence-Sum Relationships

Finding nth term from the Sum

Let \( S = (s_n) \) be a series generated by a sequence \( T = (t_n) \) , where each term \( s_n \) of the series \( S \) is defined as the sum of the first \( n \) terms of the sequence \( T \). Then \( s_1 = t_1 \), and for each \( n > 1 \), \( t_n = s_n - s_{n-1} \).

Proof:

Given a sequence \( T = (t_n) \), a series \( S = (s_n) \) is constructed such that each term \( s_n \) is the sum of the first \( n \) terms of \( T \), i.e.,

\[ s_n = t_1 + t_2 + \ldots + t_n \quad \text{for all} \quad n \in \mathbb{N} \]

By definition of the series:

  1. For \( n = 1 \), the first partial sum \( s_1 \) is defined to be equal to the first term of the sequence \( T \), which gives us:
\[ s_1 = t_1 \]

This establishes the base case of our theorem.

  1. For \( n > 1 \), consider the \( n \)-th partial sum \( s_n \) and the \( (n-1) \)-th partial sum \( s_{n-1} \):
\[ s_n = t_1 + t_2 + \ldots + t_{n-1} + t_n \]
\[ s_{n-1} = t_1 + t_2 + \ldots + t_{n-1} \]

Subtracting \( s_{n-1} \) from \( s_n \), we have:

\[ s_n - s_{n-1} = (t_1 + t_2 + \ldots + t_{n-1} + t_n) - (t_1 + t_2 + \ldots + t_{n-1}) \]
\[ s_n - s_{n-1} = t_n \]

Thus, for each \( n > 1 \), it is shown that \( t_n \), the \( n \)-th term of the sequence \( T \), is equal to the difference between the \( n \)-th and \( (n-1) \)-th partial sums of the series \( S \).

Hence, the theorem is proven.

Example

Problem:

Given a sequence \( (a_n) \) with the property that its partial sum up to \( n \) terms, denoted by \( S_n \), is \( S_n = n^2 \), find the \( n \)-th term \( a_n \) of the sequence.

Solution:

We are given that the partial sum \( S_n = n^2 \) for the sequence \( (a_n) \). We also know the formula for finding the \( n \)-th term of the sequence is \( a_n = S_n - S_{n-1} \).

  1. Calculate \( S_n \): From the given information, \( S_n = n^2 \).

  2. Calculate \( S_{n-1} \): To find \( S_{n-1} \), we substitute \( n-1 \) into the given formula for \( S_n \), giving us \( S_{n-1} = (n-1)^2 \).

  3. Find \( a_n \): Using the formula \( a_n = S_n - S_{n-1} \), we substitute the expressions for \( S_n \) and \( S_{n-1} \) into this formula:

\[ a_n = n^2 - (n-1)^2 \]

Expanding the square on the right-hand side:

\[ a_n = n^2 - (n^2 - 2n + 1) \]

Simplifying the expression:

\[ a_n = n^2 - n^2 + 2n - 1 \]
\[ a_n = 2n - 1 \]

Conclusion:

Therefore, the \( n \)-th term \( a_n \) of the sequence \( (a_n) \), where the partial sum \( S_n = n^2 \), is \( a_n = 2n - 1 \).

Sum of n consecutive terms following the first m terms

Let \( S = (s_k) \) be a series generated by a sequence \( (t_n) \). If the sum of the first \( m \) terms of the series is denoted by \( s_m \), then the sum of the next \( n \) terms is given by \( s_{m+n} - s_m \).

Proof:

By definition, the \( k \)-th partial sum \( s_k \) of the series \( S \) is the sum of the first \( k \) terms of the sequence \( (t_n) \), that is:

\[ s_k = t_1 + t_2 + \ldots + t_k \]

For \( k = m \), the sum of the first \( m \) terms is:

\[ s_m = t_1 + t_2 + \ldots + t_m \]

For \( k = m + n \), the sum of the first \( m + n \) terms is:

\[ s_{m+n} = t_1 + t_2 + \ldots + t_m + t_{m+1} + \ldots + t_{m+n} \]

To find the sum of the terms from \( t_{m+1} \) to \( t_{m+n} \), we subtract the first \( m \) terms from the sum of the first \( m + n \) terms:

\[ s_{m+n} - s_m = (t_1 + \ldots + t_m + t_{m+1} + \ldots + t_{m+n}) - (t_1 + \ldots + t_m) \]
\[ s_{m+n} - s_m = t_{m+1} + t_{m+2} + \ldots + t_{m+n} \]

Therefore, the sum of the \( n \) terms following the first \( m \) terms is \( s_{m+n} - s_m \), which completes the proof.

Increasing and Decreasing Behaviour of Sequences

Increasing Sequence

Definition: A sequence \((a_n)\) is said to be increasing if for all natural numbers \(n\), \(a_n < a_{n+1}\). This means that each term in the sequence is strictly greater than the preceding term.

Example: Consider the sequence \(a_n = n\). This is an increasing sequence because each term is exactly one more than the previous term. For instance, \(1 < 2 < 3 < \ldots\).

Decreasing Sequence

Definition: A sequence \((b_n)\) is said to be decreasing if for all natural numbers \(n\), \(b_n > b_{n+1}\). This means that each term in the sequence is strictly less than the preceding term.

Example: Consider the sequence \(b_n = \frac{1}{n}\). This is a decreasing sequence because each term is smaller than the previous term. For example, \(\frac{1}{1} > \frac{1}{2} > \frac{1}{3} > \ldots\).

Neither Increasing Nor Decreasing Sequence

Definition: A sequence \((c_n)\) is neither increasing nor decreasing if it does not strictly follow either pattern. This means there are terms in the sequence where \(c_n \leq c_{n+1}\) or \(c_n \geq c_{n+1}\) without a consistent pattern of strictly increasing or strictly decreasing values.

Example: Consider the sequence \(c_n = (-1)^n\). This sequence alternates between -1 and 1, so it is neither increasing nor decreasing. The terms of the sequence are \(1, -1, 1, -1, \ldots\), which does not show a consistent pattern of increase or decrease.

Constant Sequence

A sequence can also be monotonic without being strictly increasing or decreasing. For example, a sequence \(d_n = 2\) for all \(n\) is monotonic because it does not increase or decrease; all its terms are equal. This is an example of a constant sequence, which is a special case of a monotonic sequence.