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Practice MCQ

Unit 1 - Notes

MTH005

Unit 1: Progressions

1. Introduction to Sequences and Series

Sequence: An ordered list of numbers defined by a specific rule or function. The numbers are called terms ( $t_1, t_2, t_3, \dots, t_n$ ).
Progression: A sequence in which terms always follow a specific pattern (mathematical formula).
Series: The sum of the terms of a sequence ( $S_n = t_1 + t_2 + \dots + t_n$ ).

2. Arithmetic Progression (AP)

An Arithmetic Progression is a sequence of numbers in which the difference between any two consecutive terms is constant. This constant is known as the common difference ( $d$ ).

2.1 Standard Notation

$a$ : First term
$d$ : Common difference ( $d = t_{n} - t_{n-1}$ )
$n$ : Position of the term
$T_n$ or $a_n$ : The $n$ -th term (General Term)
$S_n$ : Sum of the first $n$ terms
$l$ : Last term

2.2 General Form

The sequence is represented as:

a, (a+d), (a+2d), (a+3d), \dots

2.3 Key Formulas

1. The $n$ -th Term ( $T_n$ )

T_n = a + (n-1)d

2. Sum of First $n$ Terms ( $S_n$ )
There are two variations of the sum formula:

When the common difference is known:
$S_n = \frac{n}{2} [2a + (n-1)d]$
When the last term ( $l$ ) is known:
$S_n = \frac{n}{2} (a + l)$

3. Arithmetic Mean (A.M.)
If three numbers $x, y, z$ are in AP, then $y$ is the Arithmetic Mean of $x$ and $z$ .

2y = x + z \quad \Rightarrow \quad y = \frac{x+z}{2}

Inserting $n$ AMs: To insert $n$ arithmetic means between two numbers $a$ and $b$ , the common difference becomes:
$d = \frac{b-a}{n+1}$

2.4 Important Properties of AP

Linearity: If a constant is added, subtracted, multiplied, or divided (non-zero) to each term of an AP, the resulting sequence is also an AP.
Symmetry:
- The sum of terms equidistant from the beginning and the end is constant and equal to the sum of the first and last terms.
- $T_k + T_{n-k+1} = a + l$
Selection of Terms: When solving problems requiring the assumption of terms in an AP, use symmetrical selection to simplify calculations:
- 3 Terms: $a-d, \ a, \ a+d$ (Sum $= 3a$ )
- 4 Terms: $a-3d, \ a-d, \ a+d, \ a+3d$ (Sum $= 4a$ ; common difference is $2d$ )
- 5 Terms: $a-2d, \ a-d, \ a, \ a+d, \ a+2d$ (Sum $= 5a$ )

3. Geometric Progression (GP)

A Geometric Progression is a sequence of non-zero numbers in which the ratio of any term to its preceding term is constant. This constant is known as the common ratio ( $r$ ).

3.1 Standard Notation

$a$ : First term
$r$ : Common ratio ( $r = \frac{t_n}{t_{n-1}}$ )
$T_n$ : The $n$ -th term

3.2 General Form

The sequence is represented as:

a, ar, ar^2, ar^3, \dots, ar^{n-1}

3.3 Key Formulas

1. The $n$ -th Term ( $T_n$ )

T_n = a \cdot r^{n-1}

2. Sum of First $n$ Terms ( $S_n$ )
The formula depends on the value of $r$ :

If $r > 1$ :
$S_n = \frac{a(r^n - 1)}{r - 1}$
If $r < 1$ :
$S_n = \frac{a(1 - r^n)}{1 - r}$
If $r = 1$ :
$S_n = n \cdot a$

3. Sum of Infinite GP ( $S_\infty$ )
This formula applies only if the GP is convergent, meaning $|r| < 1$ (i.e., $-1 < r < 1$ ).

S_\infty = \frac{a}{1 - r}

4. Geometric Mean (G.M.)
If three non-zero numbers $x, y, z$ are in GP, then $y$ is the Geometric Mean of $x$ and $z$ .

y^2 = xz \quad \Rightarrow \quad y = \sqrt{xz}

3.4 Important Properties of GP

Multiplication/Division: If every term of a GP is multiplied or divided by a non-zero constant, the resulting sequence is also a GP.
Power Rule: If every term of a GP is raised to the power $k$ , the resulting sequence is a GP with common ratio $r^k$ .
Logarithmic Relation: If $a_1, a_2, a_3, \dots$ are in GP (with positive terms), then $\log(a_1), \log(a_2), \log(a_3), \dots$ are in AP.
Selection of Terms:
- 3 Terms: $\frac{a}{r}, \ a, \ ar$ (Product $= a^3$ )
- 4 Terms: $\frac{a}{r^3}, \ \frac{a}{r}, \ ar, \ ar^3$ (Product $= a^4$ ; common ratio is $r^2$ )

4. Harmonic Progression (HP)

A sequence of non-zero numbers is a Harmonic Progression if the reciprocals of its terms form an Arithmetic Progression. Note: There is no general formula for the sum of $n$ terms of an HP.

4.1 Definition

If the sequence $a_1, a_2, a_3, \dots$ is an HP, then:

\frac{1}{a_1}, \frac{1}{a_2}, \frac{1}{a_3}, \dots \text{ is an AP}

4.2 The $n$ -th Term ( $T_n$ )

To find the $n$ -th term of an HP:

Take the reciprocals of the first two terms to find the corresponding AP.
Identify the first term ( $a$ ) and common difference ( $d$ ) of that AP.
Use the AP formula and take the reciprocal.

T_n(HP) = \frac{1}{T_n(AP)} = \frac{1}{a + (n-1)d}

(Where $a$ and $d$ belong to the corresponding AP)

4.3 Harmonic Mean (H.M.)

If $x, y, z$ are in HP, then $y$ is the Harmonic Mean of $x$ and $z$ .
Since $\frac{1}{x}, \frac{1}{y}, \frac{1}{z}$ are in AP:

\frac{1}{y} - \frac{1}{x} = \frac{1}{z} - \frac{1}{y}

\frac{2}{y} = \frac{1}{x} + \frac{1}{z} = \frac{x+z}{xz}

y = \frac{2xz}{x+z}

5. Relationships Between Means (AM, GM, HM)

For any two distinct positive numbers $a$ and $b$ :

5.1 The Definitions

Arithmetic Mean (A): $A = \frac{a+b}{2}$
Geometric Mean (G): $G = \sqrt{ab}$
Harmonic Mean (H): $H = \frac{2ab}{a+b}$

5.2 The Fundamental Identity

The square of the Geometric Mean is equal to the product of the Arithmetic and Harmonic Means.

G^2 = A \times H

5.3 The Inequality of Means

For positive real numbers (not all equal), the following inequality always holds:

AM > GM > HM

(Note: If the numbers are equal ( $a=b$ ), then $AM = GM = HM$ )

6. Summary Comparison Table

Feature	Arithmetic Progression (AP)	Geometric Progression (GP)	Harmonic Progression (HP)
Definition	Difference is constant ( $d$ )	Ratio is constant ( $r$ )	Reciprocals form an AP
Common Param	$d = T_n - T_{n-1}$	$r = T_n / T_{n-1}$	N/A (Convert to AP)
$n$ -th Term	$a + (n-1)d$	$a \cdot r^{n-1}$	$\frac{1}{a_{AP} + (n-1)d_{AP}}$
Mean ( $x, z$ )	$\frac{x+z}{2}$	$\sqrt{xz}$	$\frac{2xz}{x+z}$
Sum ( $S_n$ )	$\frac{n}{2}[2a+(n-1)d]$	$\frac{a(r^n-1)}{r-1}$	No simple general formula