6.1 Finite and infinite sequences and series

In mathematics, sequences and series are fundamental concepts used to describe ordered lists of numbers and the summation of those numbers, respectively.

Sequences

A sequence is an ordered list of numbers, where each number is called a term. Sequences can be classified as finite or infinite.

Finite Sequences

A finite sequence has a limited number of terms. It can be expressed in a list format or with a general term.

Example:
The finite sequence of the first five positive integers:
$1,2,3,4,5$

General Form:
A finite sequence can also be represented using a formula, such as an=na_n = nan​=n for $n=1,2,3,4,5$.

Infinite Sequences

An infinite sequence continues indefinitely without an end. The terms can be described using a formula or rule.

Example:
The sequence of natural numbers:
$1,2,3,4,5,\dots$

General Form:
An infinite sequence can be defined using a formula, such as an=na_n = nan​=n for $n=1,2,3,\dots$.

Series

A series is the sum of the terms of a sequence. Like sequences, series can also be finite or infinite.

Finite Series

A finite series is the sum of a finite number of terms in a sequence.

Example:
The sum of the first five positive integers can be expressed as:

$$S=1+2+3+4+5=15$$

General Form:
If ana_nan​ is the $n$-th term of a sequence, the finite series can be represented as:

Sn=a1+a2+a3+…+anS_n = a_1 + a_2 + a_3 + \ldots + a_nSn​=a1​+a2​+a3​+…+an​

Infinite Series

An infinite series is the sum of the terms of an infinite sequence.

Example:
The series formed by the infinite sequence of natural numbers:

$$S=1+2+3+4+5+\dots$$

This series diverges, meaning it does not converge to a finite limit.

Convergent and Divergent Series:

• An infinite series is convergent if the sum approaches a finite limit as more terms are added.
  Example: The geometric series S=12+14+18+…S = \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \ldotsS=21​+41​+81​+… converges to 1.
• An infinite series is divergent if the sum does not approach a finite limit.
  Example: The harmonic series S=1+12+13+…S = 1 + \frac{1}{2} + \frac{1}{3} + \ldotsS=1+21​+31​+… diverges.

6.2 Arithmetic, Geometric and Harmonic progressions

Progressions are sequences of numbers that follow a specific pattern. The three most common types are arithmetic progressions (AP), geometric progressions (GP), and harmonic progressions (HP). Each type has its own defining characteristics and formulas.

1. Arithmetic Progression (AP)

An arithmetic progression is a sequence of numbers in which the difference between consecutive terms is constant. This difference is called the common difference ($d$).

General Form

If the first term is $a$, then the $n$-th term (ana_nan​) of an arithmetic progression can be expressed as:

an=a+(n−1)da_n = a + (n-1)dan​=a+(n−1)d

Sum of the First $n$ Terms

The sum (SnS_nSn​) of the first $n$ terms of an arithmetic progression is given by:

Sn=n2(2a+(n−1)d)orSn=n2(a+an)S_n = \frac{n}{2} (2a + (n-1)d) \quad \text{or} \quad S_n = \frac{n}{2} (a + a_n)Sn​=2n​(2a+(n−1)d)orSn​=2n​(a+an​)

Example

Consider the sequence:
$2,5,8,11,14,\dots$ Here, $a=2$ and $d=3$.

• The 5th term:

  a5=2+(5−1)⋅3=2+12=14a_5 = 2 + (5-1) \cdot 3 = 2 + 12 = 14a5​=2+(5−1)⋅3=2+12=14

• The sum of the first 5 terms:

  S5=52(2⋅2+(5−1)⋅3)=52(4+12)=52⋅16=40S_5 = \frac{5}{2} (2 \cdot 2 + (5-1) \cdot 3) = \frac{5}{2} (4 + 12) = \frac{5}{2} \cdot 16 = 40S5​=25​(2⋅2+(5−1)⋅3)=25​(4+12)=25​⋅16=40

2. Geometric Progression (GP)

A geometric progression is a sequence in which each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio ($r$).

General Form

If the first term is $a$, then the $n$-th term (ana_nan​) of a geometric progression can be expressed as:

an=a⋅r(n−1)a_n = a \cdot r^{(n-1)}an​=a⋅r(n−1)

Sum of the First $n$ Terms

The sum (SnS_nSn​) of the first $n$ terms of a geometric progression is given by:

Sn=a1−rn1−rif r≠1S_n = a \frac{1 – r^n}{1 – r} \quad \text{if } r \neq 1Sn​=a1−r1−rn​if r=1

Example

Consider the sequence:
$3,6,12,24,48,\dots$ Here, $a=3$ and $r=2$.

• The 5th term:

  a5=3⋅2(5−1)=3⋅16=48a_5 = 3 \cdot 2^{(5-1)} = 3 \cdot 16 = 48a5​=3⋅2(5−1)=3⋅16=48

• The sum of the first 5 terms:

  S5=31−251−2=31−32−1=3⋅31=93S_5 = 3 \frac{1 – 2^5}{1 – 2} = 3 \frac{1 – 32}{-1} = 3 \cdot 31 = 93S5​=31−21−25​=3−11−32​=3⋅31=93

3. Harmonic Progression (HP)

A harmonic progression is a sequence of numbers whose reciprocals form an arithmetic progression.

General Form

If the terms of a harmonic progression are a1,a2,a3,…a_1, a_2, a_3, \ldotsa1​,a2​,a3​,…, then the sequence of their reciprocals:

1a1,1a2,1a3,…\frac{1}{a_1}, \frac{1}{a_2}, \frac{1}{a_3}, \ldotsa1​1​,a2​1​,a3​1​,…

is an arithmetic progression.

Example

Consider the sequence of terms:
1,12,13,14,151, \frac{1}{2}, \frac{1}{3}, \frac{1}{4}, \frac{1}{5}1,21​,31​,41​,51​ The reciprocals are:
$1,2,3,4,5$ This is an arithmetic progression with $d=1$.

Sum of the First $n$ Terms

Finding the sum of a harmonic progression is less straightforward than AP and GP. However, the sum of the first $n$ terms can be calculated using the formula for the sum of reciprocals:

Sn=nHnS_n = \frac{n}{H_n}Sn​=Hn​n​

where HnH_nHn​ is the $n$-th harmonic number:

Hn=1+12+13+…+1nH_n = 1 + \frac{1}{2} + \frac{1}{3} + \ldots + \frac{1}{n}Hn​=1+21​+31​+…+n1​

6.3 Arithmetic, Geometric, and Harmonic Means

Means are measures used to summarize a set of numbers and find a central value. The three common types of means are the arithmetic mean, geometric mean, and harmonic mean. Each has its own method of calculation and application.

1. Arithmetic Mean (AM)

The arithmetic mean is the most common type of average. It is calculated by summing a set of values and dividing by the number of values.

Formula

For a set of $n$ numbers x1,x2,…,xnx_1, x_2, \ldots, x_nx1​,x2​,…,xn​, the arithmetic mean ($AM$) is given by:

AM=x1+x2+…+xnnAM = \frac{x_1 + x_2 + \ldots + x_n}{n}AM=nx1​+x2​+…+xn​​

Example

For the numbers $4,8,12$:

AM=4+8+123=243=8AM = \frac{4 + 8 + 12}{3} = \frac{24}{3} = 8AM=34+8+12​=324​=8

2. Geometric Mean (GM)

The geometric mean is used to find the average of a set of numbers that are multiplied together. It is particularly useful for sets of positive numbers and for data that are exponentially distributed.

Formula

For a set of $n$ positive numbers x1,x2,…,xnx_1, x_2, \ldots, x_nx1​,x2​,…,xn​, the geometric mean ($GM$) is given by:

GM=x1×x2×…×xnnGM = \sqrt[n]{x_1 \times x_2 \times \ldots \times x_n}GM=nx1​×x2​×…×xn​​

Example

For the numbers $4,8,12$:

GM=4×8×123=3843≈7.937GM = \sqrt[3]{4 \times 8 \times 12} = \sqrt[3]{384} \approx 7.937GM=34×8×12​=3384​≈7.937

3. Harmonic Mean (HM)

The harmonic mean is useful for rates and ratios, particularly when dealing with quantities like speed or density. It is the reciprocal of the average of the reciprocals of the numbers.

Formula

For a set of $n$ positive numbers x1,x2,…,xnx_1, x_2, \ldots, x_nx1​,x2​,…,xn​, the harmonic mean ($HM$) is given by:

HM=n1x1+1×2+…+1xnHM = \frac{n}{\frac{1}{x_1} + \frac{1}{x_2} + \ldots + \frac{1}{x_n}}HM=x1​1​+x2​1​+…+xn​1​n​

Example

For the numbers $4,8,12$:

HM=314+18+112=30.25+0.125+0.0833=30.4583≈6.54HM = \frac{3}{\frac{1}{4} + \frac{1}{8} + \frac{1}{12}} = \frac{3}{0.25 + 0.125 + 0.0833} = \frac{3}{0.4583} \approx 6.54HM=41​+81​+121​3​=0.25+0.125+0.08333​=0.45833​≈6.54

Relationship Among the Means

For any set of positive numbers, the following inequality holds:

$$AM\ge GM\ge HM$$

This means the arithmetic mean is always greater than or equal to the geometric mean, which in turn is greater than or equal to the harmonic mean.

6.4 Sum of Arithmetic and Geometric Series

The sum of a series is the total obtained by adding the terms of the series. This section focuses on the sums of arithmetic series and geometric series, including their formulas and examples.

1. Sum of an Arithmetic Series

An arithmetic series is the sum of the terms of an arithmetic progression (AP).

Formula for the Sum of an Arithmetic Series

If the first term of the AP is $a$, the common difference is $d$, and there are $n$ terms, the sum (SnS_nSn​) can be calculated using the formula:

Sn=n2(2a+(n−1)d)S_n = \frac{n}{2} (2a + (n-1)d)Sn​=2n​(2a+(n−1)d)

Alternatively, if the last term ($l$) is known, the sum can also be calculated as:

Sn=n2(a+l)S_n = \frac{n}{2} (a + l)Sn​=2n​(a+l)

Example

Consider an arithmetic series with the first term $a=3$, a common difference $d=2$, and $n=5$ terms.

1. The terms of the series are:
   $3,5,7,9,11$.
2. The last term $l$ is:
   $l=a+(n-1)d=3+(5-1)\cdot 2=3+8=11$.
3. Calculate the sum:

   S5=52(3+11)=52⋅14=35S_5 = \frac{5}{2} (3 + 11) = \frac{5}{2} \cdot 14 = 35S5​=25​(3+11)=25​⋅14=35

2. Sum of a Geometric Series

A geometric series is the sum of the terms of a geometric progression (GP).

Formula for the Sum of a Finite Geometric Series

If the first term of the GP is $a$ and the common ratio is $r$ (where r≠1r \neq 1r=1), the sum of the first $n$ terms (SnS_nSn​) is given by:

Sn=a1−rn1−rS_n = a \frac{1 – r^n}{1 – r}Sn​=a1−r1−rn​

Example

Consider a geometric series with the first term $a=2$, a common ratio $r=3$, and $n=4$ terms.

1. The terms of the series are:
   $2,6,18,54$.
2. Calculate the sum:

   S4=21−341−3=21−81−2=2⋅−80−2=80S_4 = 2 \frac{1 – 3^4}{1 – 3} = 2 \frac{1 – 81}{-2} = 2 \cdot \frac{-80}{-2} = 80S4​=21−31−34​=2−21−81​=2⋅−2−80​=80

Sum of an Infinite Geometric Series

If $|r|<1$, the infinite geometric series converges, and the sum ($S$) is given by:

S=a1−rS = \frac{a}{1 – r}S=1−ra​

Example

For an infinite geometric series with $a=4$ and r=12r = \frac{1}{2}r=21​:

S=41−12=412=8S = \frac{4}{1 – \frac{1}{2}} = \frac{4}{\frac{1}{2}} = 8S=1−21​4​=21​4​=8

6.5 Properties of arithmetic and geometric means

Both the arithmetic mean (AM) and geometric mean (GM) have distinct properties that are useful in various mathematical contexts. Here, we’ll explore their definitions, properties, and relationships.

1. Arithmetic Mean (AM)

Definition

The arithmetic mean of a set of $n$ numbers x1,x2,…,xnx_1, x_2, \ldots, x_nx1​,x2​,…,xn​ is given by:

AM=x1+x2+…+xnnAM = \frac{x_1 + x_2 + \ldots + x_n}{n}AM=nx1​+x2​+…+xn​​

Properties of Arithmetic Mean

1. Non-negativity: The AM is non-negative if all the terms are non-negative.
2. Additive Property: If $a$ and $b$ are two groups of numbers, the AM of the combined group is the weighted average of their AMs, weighted by the number of elements in each group: AM=na⋅AMa+nb⋅AMbna+nbAM = \frac{n_a \cdot AM_a + n_b \cdot AM_b}{n_a + n_b}AM=na​+nb​na​⋅AMa​+nb​⋅AMb​​ where nan_ana​ and nbn_bnb​ are the number of terms in groups $a$ and $b$, respectively.
3. Sensitivity to Extreme Values: The AM is sensitive to extreme values (outliers). A single very high or low value can significantly affect the AM.
4. AM-GM Inequality: For any set of non-negative numbers, the AM is always greater than or equal to the GM: $$AM\ge GM$$

2. Geometric Mean (GM)

Definition

The geometric mean of a set of $n$ positive numbers x1,x2,…,xnx_1, x_2, \ldots, x_nx1​,x2​,…,xn​ is given by:

GM=x1×x2×…×xnnGM = \sqrt[n]{x_1 \times x_2 \times \ldots \times x_n}GM=nx1​×x2​×…×xn​​

Properties of Geometric Mean

1. Non-negativity: The GM is always non-negative for positive numbers.
2. Multiplicative Property: The GM of two or more groups can be found using the product of their GMs: GM=(GMa)na×(GMb)nbna+nbGM = \sqrt[n_a + n_b]{(GM_a)^{n_a} \times (GM_b)^{n_b}}GM=na​+nb​(GMa​)na​×(GMb​)nb​​
3. Less Sensitive to Extreme Values: The GM is less affected by extreme values compared to the AM. It tends to smooth out large fluctuations in the data.
4. AM-GM Inequality: The AM is always greater than or equal to the GM: $$AM\ge GM$$ This inequality holds with equality when all the numbers in the set are equal.

3. Relationship Between AM and GM

• The AM-GM inequality states that for any non-negative real numbers, the AM is always greater than or equal to the GM. This is a fundamental result in mathematics and is often used in optimization problems.
• The equality $AM=GM$ occurs if and only if all numbers in the set are identical.

6.6 Relation between means

The arithmetic mean (AM), geometric mean (GM), and harmonic mean (HM) are all measures of central tendency, but they behave differently depending on the set of values being analyzed. Understanding the relationships between these means is crucial in various mathematical applications.

1. Definitions Recap

• Arithmetic Mean (AM) of $n$ numbers x1,x2,…,xnx_1, x_2, \ldots, x_nx1​,x2​,…,xn​:

  AM=x1+x2+…+xnnAM = \frac{x_1 + x_2 + \ldots + x_n}{n}AM=nx1​+x2​+…+xn​​

• Geometric Mean (GM) of $n$ positive numbers x1,x2,…,xnx_1, x_2, \ldots, x_nx1​,x2​,…,xn​:

  GM=x1×x2×…×xnnGM = \sqrt[n]{x_1 \times x_2 \times \ldots \times x_n}GM=nx1​×x2​×…×xn​​

• Harmonic Mean (HM) of $n$ positive numbers x1,x2,…,xnx_1, x_2, \ldots, x_nx1​,x2​,…,xn​:

  HM=n1x1+1×2+…+1xnHM = \frac{n}{\frac{1}{x_1} + \frac{1}{x_2} + \ldots + \frac{1}{x_n}}HM=x1​1​+x2​1​+…+xn​1​n​

2. Relationships Among the Means

Inequalities

1. AM-GM Inequality:
   • For any set of non-negative numbers, the arithmetic mean is always greater than or equal to the geometric mean:

   $$AM\ge GM$$

   • Equality holds if and only if all numbers are equal.
2. AM-HM Inequality:
   • The arithmetic mean is also greater than or equal to the harmonic mean:

   $$AM\ge HM$$

   • This inequality holds with equality when all numbers are the same.
3. GM-HM Relationship:
   • The geometric mean is always greater than or equal to the harmonic mean:

   $$GM\ge HM$$

   • Equality occurs when all numbers are equal.

Combined Inequalities:

From the above relationships, we can summarize:

$$AM\ge GM\ge HM$$

3. Practical Implications

• Sensitivity to Values:
  • The AM is sensitive to extreme values (outliers), which can disproportionately affect the mean. This is less of an issue for the GM and HM.
• Use Cases:
  • The AM is commonly used in everyday situations (e.g., average grades, incomes), while the GM is often used in financial contexts (e.g., growth rates), and the HM is suitable for rates (e.g., speeds).