SEQUENCE

Brief Overview

graph TD; A[Sequences] --> B[Arithmetic Progression AP]; A --> C[Geometric Progression GP]; A --> D[Harmonic Progression HP]; B --> E[Constant Difference d]; C --> F[Constant Ratio r]; D --> G[Reciprocals form AP];

Introduction

A sequence is an enumerated collection of objects in which repetitions are allowed and order matters. In mathematics, we primarily deal with sequences of numbers that follow a specific pattern or rule. A series is the sum of the terms of a sequence.

1. Definition of sequence and series

Sequence: A list of numbers arranged in a definite order according to some rule. The individual numbers are called terms, denoted as $a_1, a_2, a_3, \dots, a_n$.
- Example: 2, 4, 6, 8, ... is a sequence of even numbers.
Finite vs Infinite Sequence: A sequence containing a finite number of terms is a finite sequence; otherwise, it is infinite.
Series: When the terms of a sequence are added together, the resulting expression is called a series.
- Example: If the sequence is $a_1, a_2, a_3, \dots$, then the series is $a_1 + a_2 + a_3 + \dots$

MCQs - Definition of sequence and series

A sequence is defined as a list of numbers arranged in a:
a) Random order
b) Definite order according to a rule
c) Descending order only
d) Form of a grid
Answer: b
The individual numbers in a sequence are called:
a) Factors
b) Elements or terms
c) Products
d) Roots
Answer: b
Which of the following represents a series?
a) 1, 3, 5, 7
b) $1 + 3 + 5 + 7$
c) ${1, 3, 5, 7}$
d) $1 \times 3 \times 5 \times 7$
Answer: b
A sequence that does not have a last term is called an:
a) Empty sequence
b) Infinite sequence
c) Finite sequence
d) Null sequence
Answer: b
If the $n$-th term of a sequence is $a_n = 2n + 1$, the 3rd term is:
a) 3
b) 5
c) 7
d) 9
Answer: c (Because $2(3) + 1 = 7$)
A sequence where each term is larger than the previous term is called:
a) Decreasing sequence
b) Constant sequence
c) Increasing sequence
d) Null sequence
Answer: c
The series $2 + 4 + 6 + 8$ has how many terms?
a) 2
b) 4
c) 6
d) 8
Answer: b
What is the next term in the sequence 1, 4, 9, 16, 25, ...?
a) 30
b) 36
c) 49
d) 64
Answer: b
If $a_n = n^2 - 1$, what is the 4th term?
a) 7
b) 8
c) 15
d) 16
Answer: c
A sequence is usually denoted by a set of numbers enclosed in:
a) Parentheses
b) Brackets or curly braces
c) Quotation marks
d) Straight lines
Answer: b

2. Arithmetic Progression (AP), Geometric Progression (GP), Harmonic Progression (HP)

Arithmetic Progression (AP): A sequence where the difference between consecutive terms is constant. This difference is called the common difference ($d$).
- Formula for $n$-th term: $a_n = a + (n-1)d$ (where $a$ is the first term).
- Example: 5, 8, 11, 14... ($a=5$, $d=3$).
Geometric Progression (GP): A sequence where the ratio of any term to its preceding term is constant. This is called the common ratio ($r$).
- Formula for $n$-th term: $a_n = a \cdot r^{n-1}$ (where $a$ is the first term).
- Example: 2, 6, 18, 54... ($a=2$, $r=3$).
Harmonic Progression (HP): A sequence of numbers whose reciprocals form an Arithmetic Progression.
- Example: $1, \frac{1}{2}, \frac{1}{3}, \frac{1}{4} \dots$ is an HP because $1, 2, 3, 4 \dots$ is an AP.

MCQs - AP, GP, HP

In an Arithmetic Progression, what remains constant between consecutive terms?
a) The ratio
b) The product
c) The difference
d) The sum
Answer: c
What is the common ratio ($r$) of the GP: 3, 12, 48, 192?
a) 3
b) 4
c) 9
d) 36
Answer: b (Because $12/3 = 4$, $48/12 = 4$)
Which of the following sequences is a Harmonic Progression?
a) 2, 4, 6, 8
b) $\frac{1}{2}, \frac{1}{4}, \frac{1}{8}, \frac{1}{16}$
c) $\frac{1}{3}, \frac{1}{6}, \frac{1}{9}, \frac{1}{12}$
d) 1, 4, 9, 16
Answer: c (Because the reciprocals 3, 6, 9, 12 form an AP)
The 10th term of the AP 2, 7, 12, 17... is:
a) 42
b) 47
c) 52
d) 57
Answer: b ($a_{10} = 2 + (10-1)5 = 2 + 45 = 47$)
If the first term of a GP is 5 and the common ratio is 2, the 4th term is:
a) 11
b) 20
c) 40
d) 80
Answer: c ($a_4 = 5 \times 2^{4-1} = 5 \times 8 = 40$)
What is the common difference of the AP: 10, 5, 0, -5?
a) 5
b) -5
c) 10
d) 0
Answer: b
Which of the following is an Arithmetic Progression?
a) 1, 2, 4, 8
b) 3, 6, 9, 12
c) 1, 4, 9, 16
d) 2, 3, 5, 8
Answer: b
The 5th term of a GP with $a=3$ and $r=-2$ is:
a) -48
b) 48
c) -24
d) 24
Answer: b ($a_5 = 3 \times (-2)^4 = 3 \times 16 = 48$)
In a Harmonic Progression, the reciprocals of the terms form:
a) A Geometric Progression
b) An Arithmetic Progression
c) A constant sequence
d) Another Harmonic Progression
Answer: b
If the terms $x$, $y$, $z$ are in AP, then:
a) $y = x+z$
b) $2y = x+z$
c) $y^2 = xz$
d) $y = xz$
Answer: b

3. Sum of terms in AP and GP

Sum of an AP ($S_n$): The sum of the first $n$ terms of an AP.
- Formula: $S_n = \frac{n}{2} [2a + (n-1)d]$ OR $S_n = \frac{n}{2} (a + l)$ where $l$ is the last term.
- Example: Sum of first 10 terms of 1, 3, 5... is $S_{10} = \frac{10}{2} [2(1) + (10-1)2] = 5[2 + 18] = 100$.
Sum of a finite GP ($S_n$): The sum of the first $n$ terms of a GP.
- Formula: $S_n = \frac{a(r^n - 1)}{r - 1}$ (for $r > 1$) or $S_n = \frac{a(1 - r^n)}{1 - r}$ (for $r < 1$).
Sum of an infinite GP ($S_\infty$): Only possible if $-1 < r < 1$.
- Formula: $S_\infty = \frac{a}{1 - r}$.

MCQs - Sum of terms in AP and GP

The sum of the first $n$ natural numbers (an AP with $a=1, d=1$) is given by:
a) $n(n+1)$
b) $\frac{n(n+1)}{2}$
c) $n^2$
d) $2n$
Answer: b
What is the sum of the first 5 terms of the GP: 2, 4, 8, 16, 32?
a) 60
b) 62
c) 64
d) 128
Answer: b ($S_5 = \frac{2(2^5 - 1)}{2 - 1} = 2(31) = 62$)
To calculate the sum of an infinite geometric series, the common ratio $r$ must satisfy:
a) $r > 1$
b) $r = 1$
c) $-1 < r < 1$
d) $r < -1$
Answer: c
The formula $S_n = \frac{n}{2}(a+l)$ is used to find the sum of:
a) An infinite GP
b) A finite AP when the last term is known
c) Any sequence
d) A Harmonic Progression
Answer: b
The sum of the infinite GP $1, \frac{1}{2}, \frac{1}{4}, \frac{1}{8} \dots$ is:
a) 1.5
b) 2
c) Infinity
d) 4
Answer: b ($S_\infty = \frac{1}{1 - 0.5} = \frac{1}{0.5} = 2$)
The sum of the first 100 positive integers is:
a) 5000
b) 5050
c) 10000
d) 100
Answer: b ($S_{100} = \frac{100}{2}(1 + 100) = 5050$)
For an infinite geometric series to have a finite sum, the common ratio $r$ must be:
a) $|r| > 1$
b) $|r| < 1$
c) $r = 1$
d) $r = 0$
Answer: b
The sum of the first $n$ odd natural numbers is:
a) $n(n+1)$
b) $n^2$
c) $2n$
d) $n^2 - 1$
Answer: b
What is the sum of the first 4 terms of the AP: 2, 5, 8, 11?
a) 24
b) 26
c) 28
d) 30
Answer: b
If $S_n$ represents the sum of the first $n$ terms, then the $n$-th term $a_n$ is:
a) $S_n - S_{n-1}$
b) $S_n + S_{n-1}$
c) $S_{n-1} - S_n$
d) $S_n / S_{n-1}$
Answer: a

4. Relation between Arithmetic Mean (AM), Geometric Mean (GM) and Harmonic Mean (HM)

For any two positive numbers $a$ and $b$:
* Arithmetic Mean (AM): $A = \frac{a+b}{2}$
* Geometric Mean (GM): $G = \sqrt{ab}$
* Harmonic Mean (HM): $H = \frac{2ab}{a+b}$
* Important Relationships:
1. $G^2 = A \times H$ (The square of GM equals the product of AM and HM).
2. For distinct positive numbers: $AM > GM > HM$. (If the numbers are equal, then $AM = GM = HM$).

Numerical Example:
Find AM, GM, and HM for the numbers 4 and 16.
$AM = \frac{4+16}{2} = \frac{20}{2} = 10$.
$GM = \sqrt{4 \times 16} = \sqrt{64} = 8$.
$HM = \frac{2(4)(16)}{4+16} = \frac{128}{20} = 6.4$.
Check inequality: $10 > 8 > 6.4$ ($AM > GM > HM$).
Check product: $8^2 = 64$. $A \times H = 10 \times 6.4 = 64$. They match!

MCQs - Relation between AM, GM and HM

The Arithmetic Mean of two numbers 8 and 12 is:
a) 9
b) 10
c) 96
d) 4.8
Answer: b
The Geometric Mean of two positive numbers $x$ and $y$ is:
a) $\frac{x+y}{2}$
b) $xy$
c) $\sqrt{xy}$
d) $\frac{2xy}{x+y}$
Answer: c
Which of the following inequalities correctly relates AM, GM, and HM for two distinct positive numbers?
a) $AM < GM < HM$
b) $AM = GM = HM$
c) $HM > AM > GM$
d) $AM > GM > HM$
Answer: d
What is the fundamental algebraic relationship between AM ($A$), GM ($G$), and HM ($H$)?
a) $A^2 = G \times H$
b) $G^2 = A \times H$
c) $H^2 = A \times G$
d) $A \times G \times H = 1$
Answer: b
Find the Harmonic Mean of 3 and 6.
a) 4
b) 4.5
c) $\sqrt{18}$
d) 2
Answer: a ($HM = \frac{2 \times 3 \times 6}{3+6} = \frac{36}{9} = 4$)
The Geometric Mean of 2 and 18 is:
a) 6
b) 8
c) 10
d) 36
Answer: a
For two distinct positive numbers $a$ and $b$, if $AM = 5$ and $GM = 4$, what is their $HM$?
a) 3.2
b) 3.5
c) 4.2
d) 4.5
Answer: a ($H = G^2/A = 16/5 = 3.2$)
The Arithmetic Mean of 15 and 25 is:
a) 15
b) 20
c) 25
d) 30
Answer: b
If $AM = GM$ for two positive numbers $a$ and $b$, then:
a) $a > b$
b) $a < b$
c) $a = b$
d) $a = -b$
Answer: c
The relation $AM \geq GM \geq HM$ holds true for:
a) Any real numbers
b) Only negative numbers
c) Only positive numbers
d) Complex numbers
Answer: c

Fun Facts about Sequences!

The Fibonacci Sequence: $0, 1, 1, 2, 3, 5, 8, 13 \dots$ Every number is the sum of the two preceding ones. This sequence appears everywhere in nature, from the spiral of galaxies to the arrangement of seeds on a sunflower!
Zeno's Paradox: The ancient Greek philosopher Zeno used infinite geometric series to "prove" that motion is impossible! He argued you must travel half the distance to your destination, then half of the remaining half, ad infinitum. Fortunately, calculus (and sums of infinite GPs) proves that you can reach the end!
The Golden Ratio: If you take the ratio of successive Fibonacci numbers ($1/1, 2/1, 3/2, 5/3, 8/5 \dots$), they get closer and closer to $1.6180339887\dots$, which is known as the Golden Ratio, considered the most aesthetically pleasing proportion in art and architecture.