sequence & series lec1

Introductory Remarks and Context

• Instructor Introduction: The speaker is Mohammad Hashim from the NED University Instruction Department. He is a teacher involved in preparing students for ECAT/MCAL (admission tests) particularly for NED University.

• Pedagogical Philosophy: The instructor emphasizes the difference in his teaching style compared to other mathematics teachers. He encourages students to view his numerical methods and shortcuts to understand the distinction and where he "stands" in terms of quality and efficiency.

• Current Global Situation: Mention is made of the challenges posed by the COVID-19 pandemic and the shift to online classes. The instructor expresses a first-person perspective on the depression and difficulty of learning and teaching in a pandemic context, noting that several friends and their families have been affected by the virus.

• Professional Goal: The instructor states that his greatest success is seeing his students successfully enter NED University and recognize him in the hallways.

Definitions of Sequence and Series

• Sequence: Defined as any ordered form following a particular definite rule.

  • Examples of sequences:

    • Normal numbers: $1, 2, 3, 4, 5, \dots$

    • Even numbers: $2, 4, 6, 8, 10, \dots$

    • Odd numbers, Prime numbers, or Composite numbers.

• Series: Defined as a particular sequence where the numbers follow a definite pattern and are typically linked by additive operators./

  • Examples of series:

    • $2 + 4 + 6 + 8 + 10 \dots$

    • $-2 - 4 - 6 - 8 \dots$

Types of Progressions

• The entire study of sequences and series revolves around three primary types:

  1. AP (Arithmetic Progression): A sequence formed by adding a constant value to terms.

  2. HP (Harmonic Progression): Generally understood as the reciprocal of an Arithmetic Progression.

  3. GP (Geometric Progression): A sequence formed by multiplying terms by a constant value.

Arithmetic Progression (AP)

• Definition: A sequence where every subsequent term is found by adding a specific constant number. This constant is called the Common Difference ($d$).

• Common Difference ($d$): It is the difference between any term and its previous term.

  • Formula: $d = T_2 - T_1$ or $T_n - T_{n-1}$.

  • Crucial Rule: In an AP, the value of $d$ must remain constant throughout the entire sequence. For example, in $2, 3, 4, 5, 6$, subtracting $3 - 2$ gives $1$, and $6 - 5$ also gives $1$. Therefore, $d = 1$.

• General Term (or $n^{th}$ Term):

  • Formula: $T_n = a + (n - 1)d$

  • Components:

    • $a$ : First term of the sequence.

    • $n$: Number of terms.

    • $d$: Common difference.

    • $T_n$: Can be referred to as the Last Term, the $n^{th}$ term, or the General Term.

• Standard term representations in AP:

  • First term ($T_1$): $a$

  • Second term ($T_2$): $a + d$

  • Third term ($T_3$): $a + 2d$

  • Fourth term ($T_4$): $a + 3d$

  • Sixth term ($T_6$): $a + 5d$

  • Twentieth term ($T_{20}$): $a + 19d$

  • One hundredth term ($T_{100}$): $a + 99d$

  • Generalization: Any specific term requested is always written as $a$ plus one less than the requested position multiplied by $d$.

• Variables as terms:

  • $T_p = a + (p - 1)d$

  • $T_{p+q} = a + (p + q - 1)d$

Sum of Arithmetic Progression ($S_n$)

• Primary Formula: $S_n = \frac{n}{2} [2a + (n - 1)d]$

• Shortened Formula (when the last term is known): $S_n = \frac{n}{2} [a + l]$, where $l$ is the last term ($T_n$).

• Special Case Shortcut (Sum with Middle Term): If the middle term of a series is known, then the sum can be found by:

  • Formula: $S_n = n \times \text{Middle Term}$.

  • Example: Find the sum of $17$ terms where the middle term ($9^{th}$ term) is $5$. Answer: $17 \times 5 = 85$.

• Shortcuts for Specific AP Sums:

  1. Sum of Natural Numbers ($1 + 2 + 3 + \dots + n$):

     • Formula: $\frac{n(n + 1)}{2}$

     • Example: Sum of first $20$ terms: $\frac{20 \times 21}{2} = 10 \times 21 = 210$.

  2. Sum of Odd Numbers ($1 + 3 + 5 + \dots$):

     • Formula: $n^2$

     • Example: Sum of first $20$ odd terms: $20^2 = 400$.

  3. Sum of Even Numbers ($2 + 4 + 6 + \dots$):

     • Formula: $n(n + 1)$

     • Example: Sum of first $20$ even terms: $20(20 + 1) = 20 \times 21 = 420$.

Geometric Progression (GP)

• Definition: A sequence where each term is found by multiplying the previous term by a fixed, non-zero number called the Common Ratio ($r$).

• Common Ratio ($r$): Found by dividing a term by its predecessor.

  • Formula: $r = \frac{T_2}{T_1}$.

  • Crucial Rule: The value of $r$ must be constant throughout the sequence for it to qualify as a GP.

• General Term Formula: $T_n = ar^{n-1}$

• Standard term representations in GP:

  • $T_1 = a$

  • $T_2 = ar$

  • $T_3 = ar^2$

  • $T_4 = ar^3$

  • $T_{15} = ar^{14}$

  • $T_{30} = ar^{29}$

• Variables in GP terms:

  • $T_p = ar^{p-1}$

  • $T_{p+q} = ar^{p+q-1}$

• Sum of Geometric Progression ($S_n$):

  1. When r > 1: $S_n = \frac{a(r^n - 1)}{r - 1}$

  2. When r < 1: $S_n = \frac{a(1 - r^n)}{1 - r}$

• Sum of Infinite Geometric Series ($S_{\infty}$):

  • Formula: $S_{\infty} = \frac{a}{1 - r}$

  1. Used when the number of terms is infinite and the ratio is such that the series converges.

Harmonic Progression (HP) and Reciprocals

• The instructor notes that HP does not have its own specific complex formula set for term calculation in this context. Instead, HP is the reciprocal of AP.

• Any formula valid for AP can be applied to HP by taking the reciprocal of the values.

Arithmetic and Geometric Means

• Single Arithmetic Mean (AM): The value inserted between two numbers $a$ and $b$ so that the resulting sequence is an AP.

  • Formula: $A = \frac{a + b}{2}$

• Multiple Arithmetic Means ($A_1, A_2, A_3, \dots, A_n$):

  • General Term: $A_n = a + nd$

  • Difference Formula for means: $d = \frac{b - a}{n + 1}$

  • Note: In AP, the $5^{th}$ term is $a+4d$, but the $5^{th}$ Arithmetic Mean is $a+5d$. This specific distinction is vital.

• Single Geometric Mean (GM):

  • Formula: $G = \pm \sqrt{ab}$

• Multiple Geometric Means ($G_1, G_2, G_3, \dots, G_n$):

  • General Term: $G_n = ar^n$

  • Ratio Formula for means: $r = (\frac{b}{a})^{\frac{1}{n+1}}$

• Single Harmonic Mean (HM):

  • Formula: $H = \frac{2ab}{a + b}$

Problem Solving Techniques and Case Studies

• Solving for $d$ and $a$ in AP: When given two terms like $T_4 = 7$ and $T_{16} = 31$.

  • Method: Subtract the equations. $(a + 15d) - (a + 3d) = 31 - 7 \rightarrow 12d = 24 \rightarrow d = 2$. Then substitute $d$ back to find $a = 1$.

• Solving for $n$ in Sequences: When given a general term like $T_n = n^2 - 15n$ and a value (e.g., $51$).

  • Set up a quadratic equation: $n^2 - 15n = 51 \rightarrow n^2 - 15n - 51 = 0$.

  • Use Mid-term Breaking (Factorization) or quadratic formula.

  • Philosophical/Logical Constraint on $n$: The value of $n$ (number of terms) must always be a positive integer. It cannot be negative and it cannot be a fraction/ratio.

  • Metaphor: The instructor asks how many students are in a class. One might say $40$ or $50$, but never $-3$ students or $2.5$ students. Similarly, $n$ must represent a whole, positive count.

• Solving for variables in GP sequences: If three terms are given in GP, such as $(x + 10), (x - 2), (x - 10)$.

  • Method: Since the common ratio ($r$) is equal throughout, set ratios equal: $\frac{x - 2}{x + 10} = \frac{x - 10}{x - 2}$.

  • Then cross-multiply to solve for $x$. $(x - 2)^2 = (x + 10)(x - 10) \rightarrow x^2 - 4x + 4 = x^2 - 100 \rightarrow -4x = -104 \rightarrow x = 26$.

Questions & Discussion

• Question: Why do we use certain shortcuts for ECAT preparation?

• Response: The instructor explains that time management is essential for university admission tests. While standard methods from intermediate textbooks will yield the correct answer eventually, they might take $5$ minutes. Short tricks allow students to solve problems in $20$ to $30$ seconds, which is the required pace for competitive exams.

• Question: How do we differentiate between AP and GP when it is not explicitly stated?

• Response: Check for the common difference first by subtracting terms. If the values differ, check the common ratio by dividing terms. If the ratio remains constant, it is a GP.

• Note on Student Error: Students often confuse the indexing of Arithmetic Means ($A_n = a + nd$) with standard AP terms ($T_n = a + (n-1)d$). The instructor warns to listen carefully and organize notes properly to avoid this trap.