Chapter 5 Sequences and Series (Concepts)

Sequence and Series

In mathematics, we often deal with ordered collections of numbers. These ordered collections, especially when they follow a discernible pattern or rule, are central to the concepts of Sequences and Series. These ideas are fundamental and have extensive applications in various fields, including algebra, calculus, physics, finance, and computer science.

Sequences

A Sequence is an ordered list of numbers or other mathematical objects. The key characteristic of a sequence is that the order of the elements matters. Each number in the sequence is called a term or an element of the sequence. The terms are usually denoted by using subscripts to indicate their position in the list.

A sequence is typically represented as $a_1, a_2, a_3, ..., a_n, ...$

• $a_1$ is the first term.
• $a_2$ is the second term.
• $a_3$ is the third term.
• ...
• $a_n$ is the $n$-th term, also known as the general term or formula term of the sequence.

The terms of a sequence often follow a specific pattern or rule, which can sometimes be expressed as a formula for the $n$-th term ($a_n$) in terms of its position $n$. If such a formula exists, it allows us to find any term of the sequence by simply substituting the desired position number for $n$.

Types of Sequences based on Number of Terms

• Finite Sequence: A sequence that has a limited or fixed number of terms. The list of terms has a definite beginning and a definite end.

  Example: The sequence of odd numbers less than 10: 1, 3, 5, 7, 9.

  Example: The list of ranks of students in a class in a test: 1st, 2nd, 3rd, ..., up to the last rank.

• Infinite Sequence: A sequence that has an unlimited number of terms. The list of terms continues indefinitely.

  Example: The sequence of all positive even numbers: 2, 4, 6, 8, 10, ... (The ellipsis '...' indicates that the sequence continues without end).

  Example: The sequence of reciprocals of natural numbers: $1, \frac{1}{2}, \frac{1}{3}, \frac{1}{4}, ...$

Representing Sequences

Sequences can be represented in several ways, depending on how the terms are generated or described:

• Listing the Terms (Roster Method): Writing out the first few terms of the sequence. This is effective when the pattern is clear from the initial terms, or for finite sequences where all terms can be listed. For infinite sequences with a clear pattern, an ellipsis (...) is used after the initial terms.

  Example: 1, 4, 9, 16, ... (It is clear that the $n$-th term is $n^2$)

  Example: 5, 10, 15, 20 (A finite sequence)

• Using a Formula for the General Term ($a_n$): Providing an explicit expression for the $n$-th term $a_n$ as a function of $n$. This formula allows us to find any term of the sequence directly by substituting the position number $n$.

  Example: Consider the sequence defined by $a_n = 2n$. The terms are generated by substituting $n=1, 2, 3, ...$
  For $n=1, a_1 = 2(1) = 2$.
  For $n=2, a_2 = 2(2) = 4$.
  For $n=3, a_3 = 2(3) = 6$.
  The sequence is 2, 4, 6, 8, ...

  Example: Consider the sequence defined by $a_n = (-1)^{n-1} \frac{1}{n}$.
  For $n=1, a_1 = (-1)^{1-1} \frac{1}{1} = (-1)^0 \times 1 = 1 \times 1 = 1$.
  For $n=2, a_2 = (-1)^{2-1} \frac{1}{2} = (-1)^1 \times \frac{1}{2} = - \frac{1}{2}$.
  For $n=3, a_3 = (-1)^{3-1} \frac{1}{3} = (-1)^2 \times \frac{1}{3} = 1 \times \frac{1}{3} = \frac{1}{3}$.
  The sequence is $1, -\frac{1}{2}, \frac{1}{3}, -\frac{1}{4}, ...$

• Using a Recurrence Relation (Recursive Definition): Defining a term in the sequence based on the value(s) of one or more preceding terms. This method requires specifying the value(s) of the initial term(s) to start the sequence.

  Example: The Fibonacci sequence is defined by $a_n = a_{n-1} + a_{n-2}$ for $n \geq 3$, with initial conditions $a_1 = 1$ and $a_2 = 1$.
  To find the terms:
  $a_1 = 1$
  $a_2 = 1$
  $a_3 = a_2 + a_1 = 1 + 1 = 2$.
  $a_4 = a_3 + a_2 = 2 + 1 = 3$.
  $a_5 = a_4 + a_3 = 3 + 2 = 5$.
  The sequence is 1, 1, 2, 3, 5, 8, 13, ...

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Series

A Series is simply the sum of the terms of a sequence. If we have a sequence of numbers $a_1, a_2, a_3, ..., a_n, ...$, the corresponding series is the expression obtained by adding these terms together: $a_1 + a_2 + a_3 + ... + a_n + ...$.

Series are conveniently represented using the Greek letter Sigma ($\sum$), which denotes summation. The sum of the first $n$ terms of a sequence $\{a_k\}$ is denoted by $S_n$.

Here, $\sum\limits_{k=1}^{n} a_k$ means "the sum of $a_k$ for values of $k$ starting from 1 up to $n$". The index $k$ is a dummy variable and can be any letter (e.g., $\sum\limits_{i=1}^{n} a_i$).

Types of Series

Based on whether the corresponding sequence is finite or infinite:

• Finite Series: A series formed by summing the terms of a finite sequence. A finite series always has a definite sum (a single numerical value).

  Example: $1 + 3 + 5 + 7 + 9$. This is the sum of the finite sequence 1, 3, 5, 7, 9. The sum is $1+3+5+7+9 = 25$.

• Infinite Series: A series formed by summing the terms of an infinite sequence. Represented as $a_1 + a_2 + a_3 + ...$, or using summation notation as $\sum\limits_{k=1}^{\infty} a_k$. The sum of an infinite series may converge to a finite value or diverge (not approach a finite value). The concept of convergence is a significant part of higher mathematics.

  Example: $1 + \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + ...$ (The sum here approaches 2, so it converges)

  Example: $1 + 2 + 3 + 4 + ...$ (The sum grows infinitely large, so it diverges)

In this chapter, we will primarily study specific types of sequences that follow simple patterns (Arithmetic Progression and Geometric Progression) and calculate the sums of their finite versions.

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Key Distinction between Sequence and Series

It is important to clearly differentiate between a sequence and a series:

• A Sequence is an ordered list of terms (e.g., 2, 4, 6, 8). It focuses on the individual terms and their order.
• A Series is the sum of the terms of a sequence (e.g., $2 + 4 + 6 + 8$). It focuses on the combined value obtained by adding the terms.

Think of a sequence as a collection of numbers laid out in order, and a series as the result of adding those numbers together.

Arithmetic Progression

An Arithmetic Progression (AP) is a specific type of sequence in which the difference between any term and its preceding term is constant throughout the sequence. This constant difference is called the common difference and is typically denoted by the letter $d$.

Definition and General Form of an AP

A sequence $a_1, a_2, a_3, ..., a_n, ...$ is said to be an Arithmetic Progression if $a_{n+1} - a_n = d$ for all positive integers $n$, where $d$ is a constant.

If the first term of an AP is denoted by $a$ (or $a_1$), and the common difference is $d$, then the terms of the AP can be generated sequentially:

• First term ($a_1$) $= a$
• Second term ($a_2$) $= a_1 + d = a + d$
• Third term ($a_3$) $= a_2 + d = (a + d) + d = a + 2d$
• Fourth term ($a_4$) $= a_3 + d = (a + 2d) + d = a + 3d$
• ... and so on.

Thus, the general form of an AP is:

The common difference $d$ can be any real number: positive (increasing AP), negative (decreasing AP), or zero (constant AP).

Identifying an AP

To check if a given sequence is an AP, calculate the difference between consecutive terms. If $a_2-a_1 = a_3-a_2 = a_4-a_3 = ... = d$, then the sequence is an AP with common difference $d$.

Example: The sequence 5, 10, 15, 20, 25, ...

$10-5 = 5$

$15-10 = 5$

$20-15 = 5$

$25-20 = 5$

Since the difference is constantly 5, it is an AP with first term $a=5$ and common difference $d=5$.

Example: The sequence 20, 17, 14, 11, 8, ...

$17-20 = -3$

$14-17 = -3$

$11-14 = -3$

$8-11 = -3$

Since the difference is constantly -3, it is an AP with first term $a=20$ and common difference $d=-3$.

Example: The sequence $\frac{1}{2}, 1, \frac{3}{2}, 2, \frac{5}{2}, ...$

$1 - \frac{1}{2} = \frac{1}{2}$

$\frac{3}{2} - 1 = \frac{3}{2} - \frac{2}{2} = \frac{1}{2}$

$2 - \frac{3}{2} = \frac{4}{2} - \frac{3}{2} = \frac{1}{2}$

$\frac{5}{2} - 2 = \frac{5}{2} - \frac{4}{2} = \frac{1}{2}$

Since the difference is constantly $\frac{1}{2}$, it is an AP with first term $a=\frac{1}{2}$ and common difference $d=\frac{1}{2}$.

nth Term (General Term) of an AP

The formula for the $n$-th term of an AP, denoted by $a_n$, provides a direct way to calculate any term in the sequence without having to list all the preceding terms.

Derivation of the Formula for $a_n$:

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Sum of the First n Terms of an AP

The sum of the first $n$ terms of an Arithmetic Progression is denoted by $S_n$.

Derivation of the Formula for $S_n$:

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Properties of Arithmetic Progression

Arithmetic Progressions have several useful properties:

• If a constant quantity is added to or subtracted from each term of an AP, the resulting sequence is also an AP with the same common difference.

  Example: If 1, 3, 5, 7 is an AP (d=2), then adding 10 to each term gives 11, 13, 15, 17, which is also an AP (d=2).

• If each term of an AP is multiplied or divided by a non-zero constant, the resulting sequence is also an AP. If the original AP had a common difference $d$, the new AP will have a common difference $kd$ (if multiplied by $k$) or $d/k$ (if divided by $k$).

  Example: If 2, 4, 6, 8 is an AP (d=2), then multiplying by 3 gives 6, 12, 18, 24, which is an AP with common difference $3 \times 2 = 6$.

• In a finite AP, the sum of terms equidistant from the beginning and the end is constant and is equal to the sum of the first and last terms. For an AP $a_1, a_2, ..., a_n$, the sum $a_k + a_{n-k+1}$ is always equal to $a_1 + a_n$ for $k=1, 2, ..., n$.

  Example: In the AP 2, 5, 8, 11, 14, 17 ($n=6$). $a_1=2, a_6=17$. $a_1+a_6 = 19$.
  $k=2$: $a_2=5, a_{6-2+1}=a_5=14$. $a_2+a_5 = 5+14=19$.
  $k=3$: $a_3=8, a_{6-3+1}=a_4=11$. $a_3+a_4 = 8+11=19$.

• If three numbers $a, b, c$ are in AP, then the middle term $b$ is the arithmetic mean of the other two terms. $b = \frac{a+c}{2}$, which implies $2b = a+c$.

Arithmetic Mean (AM)

When we insert numbers between two terms of an AP such that the entire sequence remains in AP, the inserted numbers are called Arithmetic Means. If $A_1, A_2, ..., A_m$ are $m$ numbers inserted between $a$ and $b$ such that $a, A_1, A_2, ..., A_m, b$ form an AP, then these $A_i$ are $m$ arithmetic means between $a$ and $b$.

If we insert a single number $A$ between $a$ and $b$ such that $a, A, b$ are in AP, then $A = \frac{a+b}{2}$. This is the simple average we discussed earlier, also known as the single arithmetic mean between $a$ and $b$.

If $a$ and $b$ are two numbers, and $d$ is the common difference of the AP $a, A_1, ..., A_m, b$, then $b$ is the $(m+2)$-th term. $b = a_{(m+2)} = a + (m+2-1)d = a + (m+1)d$. So, $b-a = (m+1)d \implies d = \frac{b-a}{m+1}$. The $m$ arithmetic means are $A_1 = a+d, A_2 = a+2d, ..., A_m = a+md$.

Geometric Progression

A Geometric Progression (GP) is another important type of sequence, characterized by a constant ratio between consecutive terms. Unlike an AP where a fixed value is added, in a GP, a fixed value is multiplied to get the next term. This constant value is called the common ratio, denoted by the letter $r$.

Definition and General Form of a GP

A sequence of non-zero numbers $a_1, a_2, a_3, ..., a_n, ...$ is said to be a Geometric Progression if the ratio of any term to its preceding term is constant for all positive integers $n$.

Mathematically, this means $\frac{a_{n+1}}{a_n} = r$ for all $n \geq 1$, where $r$ is a constant and $r \neq 0$.

If the first term of a GP is denoted by $a$ (or $a_1$), and the common ratio is $r$, then the terms of the GP can be generated sequentially by multiplying the previous term by $r$:

• First term ($a_1$) $= a$
• Second term ($a_2$) $= a_1 \times r = a \times r = ar$
• Third term ($a_3$) $= a_2 \times r = (ar) \times r = ar^2$
• Fourth term ($a_4$) $= a_3 \times r = (ar^2) \times r = ar^3$
• ... and so on.

Thus, the general form of a GP is:

It is important that all terms in a GP are non-zero. If the first term $a=0$, all terms would be 0, which trivializes the concept of a ratio. If the common ratio $r=0$, all terms after the first would be 0, which also loses the characteristic pattern unless specified as a special case. Therefore, we typically assume $a \neq 0$ and $r \neq 0$ for a GP.

The common ratio $r$ can be positive or negative, leading to different types of GPs (e.g., terms alternating in sign if $r$ is negative).

Identifying a GP

To check if a given sequence of non-zero numbers is a GP, calculate the ratio of each term to its preceding term. If this ratio is constant throughout the sequence, it is a GP with that constant ratio $r$.

Example: The sequence 2, 6, 18, 54, 162, ...

$\frac{6}{2} = 3$

$\frac{18}{6} = 3$

$\frac{54}{18} = 3$

$\frac{162}{54} = 3$

Since the ratio is constantly 3, it is a GP with first term $a=2$ and common ratio $r=3$.

Example: The sequence 16, -8, 4, -2, 1, ...

$\frac{-8}{16} = -\frac{1}{2}$

$\frac{4}{-8} = -\frac{1}{2}$

$\frac{-2}{4} = -\frac{1}{2}$

$\frac{1}{-2} = -\frac{1}{2}$

Since the ratio is constantly $-\frac{1}{2}$, it is a GP with first term $a=16$ and common ratio $r=-\frac{1}{2}$. The terms alternate in sign.

nth Term (General Term) of a GP

The formula for the $n$-th term of a GP, denoted by $a_n$, provides a direct method to calculate any term in the sequence using its position number $n$, the first term $a$, and the common ratio $r$.

Derivation of the Formula for $a_n$:

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Sum of the First n Terms of a GP

The sum of the first $n$ terms of a Geometric Progression is denoted by $S_n$.

Derivation of the Formula for $S_n$:

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Sum of Infinite Terms of a GP ($S_\infty$)

For an infinite Geometric Progression ($a + ar + ar^2 + ar^3 + ...$), the sum of all infinite terms, denoted by $S_\infty$, exists and is a finite value only if the absolute value of the common ratio $r$ is strictly less than 1 (i.e., $|r| < 1$ or, equivalently, $-1 < r < 1$).

If $|r| < 1$, as the number of terms $n$ approaches infinity ($n \to \infty$), the term $r^n$ approaches 0 ($r^n \to 0$). Looking at the formula for $S_n = \frac{a(1 - r^n)}{1 - r}$, as $n$ becomes very large, $r^n$ becomes very small and approaches 0.

Since $\lim\limits_{n \to \infty} r^n = 0$ when $|r| < 1$:

This formula gives the sum of an infinite GP when it converges (i.e., when $|r| < 1$).

If $|r| \geq 1$ (and $r \neq 1$), the terms of the GP do not approach zero, and the sum of the infinite series grows infinitely large or oscillates. In this case, the infinite series diverges and does not have a finite sum. (If $r=1$, the sum is infinite unless $a=0$).

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Properties of Geometric Progression

Geometric Progressions have several useful properties:

• If each term of a GP is multiplied or divided by a non-zero constant, the resulting sequence is also a GP with the same common ratio.

  Example: If 2, 6, 18, ... is a GP (a=2, r=3), then multiplying by 5 gives 10, 30, 90, ..., which is a GP (a=10, r=3).

• The reciprocals of the terms of a GP also form a GP. If the original GP has common ratio $r$, the sequence of reciprocals will have a common ratio $1/r$.

  Example: If 2, 4, 8, 16 is a GP (r=2), then the reciprocals are $\frac{1}{2}, \frac{1}{4}, \frac{1}{8}, \frac{1}{16}$, which form a GP with common ratio $\frac{1/4}{1/2} = \frac{1}{2} = 1/2$.

• If three non-zero numbers $a, b, c$ are in GP, then the middle term $b$ is the geometric mean of the other two terms. $b^2 = ac$.

  Example: 3, 6, 12 are in GP. $6^2 = 36$, $3 \times 12 = 36$. So $6^2 = 3 \times 12$.

• If $a_1, a_2, ..., a_n$ is a GP with positive terms, then $\log a_1, \log a_2, ..., \log a_n$ is an AP.

  Explanation: Let $a_k = ar^{k-1}$. Then $\log a_k = \log(ar^{k-1}) = \log a + \log(r^{k-1}) = \log a + (k-1)\log r$. This term is in the form $A + (k-1)D$, where $A = \log a$ and $D = \log r$. This is the general term of an AP with first term $\log a$ and common difference $\log r$.

Geometric Mean (GM)

When we insert numbers between two terms of a GP such that the entire sequence remains in GP, the inserted numbers are called Geometric Means. If $G_1, G_2, ..., G_m$ are $m$ numbers inserted between $a$ and $b$ such that $a, G_1, G_2, ..., G_m, b$ form a GP, then these $G_i$ are $m$ geometric means between $a$ and $b$.

If we insert a single number $G$ between $a$ and $b$ such that $a, G, b$ are in GP, then $\frac{G}{a} = \frac{b}{G}$, which implies $G^2 = ab$. For positive $a$ and $b$, $G = \sqrt{ab}$. This is the geometric mean of $a$ and $b$.

If $a$ and $b$ are two numbers, and $r$ is the common ratio of the GP $a, G_1, ..., G_m, b$, then $b$ is the $(m+2)$-th term. $b = a_{(m+2)} = ar^{m+2-1} = ar^{m+1}$. So, $\frac{b}{a} = r^{m+1} \implies r = \left(\frac{b}{a}\right)^{\frac{1}{m+1}}$. The $m$ geometric means are $G_1 = ar, G_2 = ar^2, ..., G_m = ar^m$.

Applications of AP and GP

Arithmetic Progressions (AP) and Geometric Progressions (GP) are not merely theoretical constructs studied in mathematics classrooms. They are powerful tools for modeling and analyzing various phenomena in the real world. Understanding how quantities change additively (AP) or multiplicatively (GP) allows us to solve problems in finance, biology, physics, economics, and many other fields.

Applications of Arithmetic Progression (AP)

Arithmetic Progression is applicable in situations where a quantity increases or decreases by a fixed amount in each successive step, time period, or iteration.

• Simple Interest: Simple interest is calculated only on the initial principal amount. This means the amount of interest earned remains constant for each interest period (e.g., each year). If this constant interest amount is added to the principal at the end of each period, the total amount (Principal + accumulated Simple Interest) at the end of successive periods forms an AP.

  Let Principal amount be $P$.

  Let Rate of simple interest be $R$ per annum (as a percentage).

  Simple Interest for 1 year $= \frac{P \times R \times 1}{100} = \frac{PR}{100}$.

  Amount at the beginning (Year 0) $= P$.

  Amount after 1 year $= P + \frac{PR}{100}$.

  Amount after 2 years $= P + \frac{PR}{100} + \frac{PR}{100} = P + 2 \left(\frac{PR}{100}\right)$.

  Amount after 3 years $= P + 3 \left(\frac{PR}{100}\right)$.

  Amount after $n$ years $= P + n \left(\frac{PR}{100}\right)$.

  The sequence of amounts at the end of Year 0, Year 1, Year 2, ... is $P, P + \frac{PR}{100}, P + 2\frac{PR}{100}, P + 3\frac{PR}{100}, ..., P + n\frac{PR}{100}$.

  This sequence is an AP with the first term (Year 0) as $P$ and the common difference $d = \frac{PR}{100}$. The amount after $n$ years is the $(n+1)$-th term of this AP (starting from the 0th term $a_0=P$) or it's the $n$-th term of the sequence of *amounts accumulated* starting from Year 1, $a_n = (P + \frac{PR}{100}) + (n-1)\frac{PR}{100} = P + \frac{PR}{100} + n\frac{PR}{100} - \frac{PR}{100} = P + n\frac{PR}{100}$.

• Linear Growth or Decay: Many natural or artificial processes involve a quantity changing by a fixed amount per unit of time or per step. Examples include:

  • The height of a plant increasing by a constant amount each week.
  • The distance covered by an object moving with constant acceleration (displacement in equal time intervals).
  • Depreciation of an asset by a fixed amount each year (straight-line depreciation).
  • The balance in a bank account where a fixed amount is deposited or withdrawn regularly (ignoring interest).

• Equal Instalments: In some financial arrangements like loan repayments or recurring deposits with fixed principal reduction/addition, the series of payments or balances might follow an AP or relate to the sum of an AP.

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Applications of Geometric Progression (GP)

Geometric Progression is applicable in situations where a quantity increases or decreases by a constant percentage or factor in each successive step, time period, or iteration.

• Compound Interest: Compound interest is calculated on the principal amount as well as the accumulated interest from previous periods. This leads to the amount growing by a constant multiplication factor each period. The amounts at the end of successive periods form a GP.

  Let Principal amount be $P$.

  Let Rate of compound interest be $R$ per annum (as a percentage), compounded annually.

  The growth factor per year is $(1 + \frac{R}{100})$. Let $r = 1 + \frac{R}{100}$.

  Amount at the beginning (Year 0) $= P$. (We can consider this as $a_0$).

  Amount after 1 year $= P \times r = Pr$. (This is $a_1$ if $P$ is $a_0$, or $a_1$ if we start sequence from Year 1 amount).

  Amount after 2 years $= (Pr) \times r = Pr^2$.

  Amount after 3 years $= (Pr^2) \times r = Pr^3$.

  Amount after $n$ years $= Pr^n = P \left(1 + \frac{R}{100}\right)^n$.

  The sequence of amounts at the end of Year 0, Year 1, Year 2, ... is $P, P(1+\frac{R}{100}), P(1+\frac{R}{100})^2, ..., P(1+\frac{R}{100})^n$. This is a GP with first term $P$ and common ratio $(1+\frac{R}{100})$. The amount after $n$ years is the $(n+1)$-th term ($a_n$) of this sequence (if we start counting terms from $n=0$ for the initial principal) or the $n$-th term of the sequence of amounts starting from Year 1 as the first term ($a_1$).

• Population Growth or Decay: If a population of people, animals, or bacteria increases or decreases by a fixed percentage each year or generation, the population size over time can be modeled using a GP.

• Depreciation: If the value of an asset (like machinery, vehicles) decreases by a fixed percentage each year (reducing balance method of depreciation), its value over successive years forms a GP.

• Radioactive Decay: The amount of a radioactive substance decreases by a fixed percentage over equal periods (half-life concept is related to GP with $r = 1/2$).

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