Arithmetic Progressions Lecture Notes

Vocabulary flashcards covering definitions, formulas, and examples of Arithmetic Progressions based on the lecture transcript.

Term

Each of the numbers in a list of numbers.

Arithmetic Progression (AP)

A list of numbers in which each term is obtained by adding a fixed number to the preceding term except the first term.

Common Difference

The fixed number added to each preceding term in an AP, denoted by $d$. It can be positive, negative, or zero.

First Term

The starting number in an arithmetic progression, usually denoted by $a$ or $a_1$.

General Form of an AP

Represented as $a, a + d, a + 2d, a + 3d, \dots$ where $a$ is the first term and $d$ is the common difference.

Finite AP

An arithmetic progression that contains only a finite number of terms and always has a last term.

Infinite AP

An arithmetic progression that is not finite and does not have a last term.

Last Term

The final term in a finite arithmetic progression, often denoted by $l$ or $a_n$.

$n^{th}$ Term Formula

The formula given by $a_n = a + (n - 1)d$ to find any specific term in an AP.

General Term

Another name for the $n^{th}$ term ($a_n$) of an arithmetic progression.

Sum of First $n$ Terms ($S_n$) Standard Formula

The sum calculated using the formula $S = \frac{n}{2}[2a + (n - 1)d]$.

Sum of First $n$ Terms ($S_n$) with Last Term

The sum calculated using the formula $S = \frac{n}{2}(a + l)$ where $l$ is the last term.

Arithmetic Mean

If $a, b, c$ are in AP, then $b = \frac{a + c}{2}$ is the middle term.

Common Difference Equation

In general, for an AP $a_1, a_2, \dots, a_n$, it is defined as $d = a_{k+1} - a_k$ for any valid integer $k$.

Relationship between $a_n$ and $S_n$

The $n^{th}$ term of an AP is the difference between the sum of first $n$ terms and the sum of first $(n - 1)$ terms, expressed as $a_n = S_n - S_{n-1}$.

Sum of First $n$ Positive Integers

The mathematical sum given by the formula $S_n = \frac{n(n + 1)}{2}$.

Reena's Salary Pattern

A real-life AP example starting at $8000$ with an annual increment of $500$, resulting in $8000, 8500, 9000, \dots$.

Ladder Rungs Pattern

A problem where rung lengths decrease uniformly by $2\,cm$ from bottom to top, starting from $45\,cm$.

Succeeding Term

The term that immediately follows a given term in a sequence.

Preceding Term

The term that immediately comes before a given term in a sequence.

Unit Square Pattern

A sequence of unit squares with sides $1, 2, 3, \dots$ resulting in square totals $1^2, 2^2, 3^2, \dots$, which is not an AP.

Shakila's Daughter's Money Box

An AP example where the initial amount is $100$ and increases by $50$ every year.

Rabbit Pair Sequence

A pattern of rabbit pairs represented by $1, 1, 2, 3, 5, 8$, which does not form an AP.

Common Difference of $100, 70, 40, 10, \dots$

The fixed number is $-30$.

Common Difference of $-1.0, -1.5, -2.0, -2.5, \dots$

The fixed number is $-0.5$.

Common Difference of $3, 3, 3, 3, \dots$

The fixed number is $0$.

Finite AP Example: Assembly Queue

Heights of students in a queue $(147, 148, 149, \dots, 157)$.

Finite AP Example: Cash Prizes

Rewards for Classes I to XII ranging from $200$ to $750$ with increments of $50$; $200, 250, 300, \dots, 750$.

Subba Rao's Annual Salary

Starting at $5000$ in $1995$ with an annual increment of $200$.

Simple Interest Formula in AP

$Simple\,Interest = \frac{P \times R \times T}{100}$, used to generate a sequence of interessi per year.

Positive Integer check for $n$

The value of $n$ in an AP must always be a positive integer because it represents the position of a term.

Rose Plant Problem

An AP scenario with $23$ plants in the first row and a common difference of $-2$, ending with $5$ plants.

Gauss's Summation Method

A technique for finding the sum of integers from $1$ to $100$ by adding the sequence to its reverse to get $101 \times 100$.

Sum of first $1000$ positive integers

The calculated value is $500500$.

Two-digit numbers divisible by 3

The sequence is $12, 15, 18, \dots, 99$, totaling $30$ terms.

Common difference of $3, 3 + \sqrt{2}, 3 + 2\sqrt{2}, \dots$

The fixed number added is $\sqrt{2}$.

Sum of first $22$ terms of $8, 3, -2, \dots$

The total sum is $-979$.

Ramkali's Weekly Savings

Starts with $5$ and increases by $1.75$ each week.

Divisibility by 7: Three-digit numbers

An AP starting at $105$ and ending at $994$.

Multiples of 4 between 10 and 250

The sequence starts at $12$ and ends at $248$, forming an AP.

Sum of terms from 5th to 13th in AP sum problem

In Example 13, this sum is zero because signs cancel out due to a positive $a$ and negative $d$.

TV Manufacturer AP

Production of $600$ sets in year 3 and $700$ sets in year 7, assuming uniform annual increase.

Simple Interest AP Example

Interest on $1000$ at $8\%$ yields the sequence $80, 160, 240, \dots$ with $d = 80$.

Construction Delay Penalty

$200$ for the first day, increasing by $50$ each subsequent day.

School Tree Planting Logic

Number of trees planted equals the class number (Class I plants 1, II plants 2) with three sections per class.

Spiral Length Formula Concept

Total length of a spiral made of $13$ consecutive semicircles with increasing radii: $0.5, 1.0, 1.5, \dots$.

Log Stacking AP

$200$ logs arranged with $20$ in the bottom row, $19$ in the next, and so on.

Potato Race Movement

Calculated as $2 \times 5 + 2 \times (5 + 3) + \dots$ representing the distance run to pick and drop potatoes.

House Numbering sum equality

Finding $x$ such that $S_{x-1} = S_{49} - S_x$ where houses are numbered $1$ to $49$.

Terrace Step Volume

Volume of concrete for the first step calculated as $\frac{1}{4} \times \frac{1}{2} \times 50\,m^3$.