Comprehensive Study Notes on Arithmetic Progressions

Patterns in Nature: Many phenomena in nature follow specific mathematical patterns, including the petals of a sunflower, the holes of a honeycomb, the grains on a maize cob, the spirals on a pineapple, and the spirals on a pine cone.
Examples of Real-Life Patterns:
- Salary Growth: Reena's starting monthly salary is $8000\,\text{INR}$ with an annual increment of $500\,\text{INR}$ . Her salary sequence for the first few years is: $8000, 8500, 9000, . . .$
- Ladder Rungs: A ladder's rungs decrease uniformly by $2\,\text{cm}$ from bottom to top. If the bottom rung is $45\,\text{cm}$ , the lengths are: $45, 43, 41, 39, 37, 35, 33, 31$
- Savings Scheme: An investment of $8000\,\text{INR}$ grows to $\frac{5}{4}$ times its value every $3\,\text{years}$ . The maturity amounts after $3, 6, 9\text{ and } 12\,\text{years}$ are: $10000, 12500, 15625, 19531.25$
- Unit Squares: The number of unit squares in larger squares with sides $1, 2, 3, . . .\,\text{units}$ is: $1^2, 2^2, 3^2, . . .$
- Money Box Savings: Shakila puts $100\,\text{INR}$ into a money box on her daughter's 1st birthday and increases the amount by $50\,\text{INR}$ every year: $100, 150, 200, 250, . . .$
- Rabbit Population (Fibonacci pattern): Pairs of rabbits that do not produce in the first month but produce a new pair every month thereafter result in the sequence: $1, 1, 2, 3, 5, 8$

Term: Each number in a list follows a certain rule and is called a term.
Definition of Arithmetic Progression (AP): An arithmetic progression is a list of numbers in which each term is obtained by adding a fixed number to the preceding term, except for the first term.
Common Difference ( $d$ ): The fixed number added to each term is called the common difference. It can be positive, negative, or zero.
Notation:
- First term: $a_1$ (or simply $a$ )
- Second term: $a_2$
- nth term: $a_n$
- Common Difference: $d$
- Relation: $a_2 - a_1 = a_3 - a_2 = . . . = a_n - a_{n-1} = d$
General Form of an AP: The sequence is represented as: $a, a + d, a + 2d, a + 3d, . . .$
Types of APs:
- Finite AP: Contains a finite number of terms and always has a last term. Examples include student heights in a queue or temperatures for a single week.
- Infinite AP: Does not have a last term and continues indefinitely. Examples include the set of natural numbers $1, 2, 3, . . .$

Derivation of the Formula:
- Let the first term be $a$ and the common difference be $d$ .
- $a_1 = a$
- $a_2 = a + d = a + (2 - 1)d$
- $a_3 = a + 2d = a + (3 - 1)d$
- $a_4 = a + 3d = a + (4 - 1)d$
General Term Formula: The nth term $a_n$ of an AP is given by: $a_n = a + (n - 1)d$
Last Term ( $l$ ): If there are $m$ terms in an AP, $a_m$ represents the last term, sometimes denoted as $l$ .
Identifying an AP: To determine if a list is an AP, calculate $d = a_{k+1} - a_k$ for various values of $k$ . If the difference is not constant, it is not an AP.

Example 3: For the AP $2, 7, 12, . . .$ , find the 10th term.
- $a = 2$
- $d = 7 - 2 = 5$
- $n = 10$
- $a_{10} = 2 + (10 - 1) \times 5 = 2 + 45 = 47$
Example 4: Which term of the AP $21, 18, 15, . . .$ is $-81$ ? Is any term $0$ ?
- $-81 = 21 + (n - 1)(-3) \rightarrow -102 = -3(n - 1) \rightarrow 34 = n - 1 \rightarrow n = 35$
- For $a_n = 0$ : $0 = 21 + (n - 1)(-3) \rightarrow -21 = -3(n - 1) \rightarrow 7 = n - 1 \rightarrow n = 8$
- Answer: The 35th term is $-81$ and the 8th term is $0$ .
Example 7: How many two-digit numbers are divisible by $3$ ?
- List: $12, 15, 18, . . . , 99$
- $a = 12, d = 3, a_n = 99$
- $99 = 12 + (n - 1)3 \rightarrow 87 = 3(n - 1) \rightarrow 29 = n - 1 \rightarrow n = 30$
Example 8: Find the 11th term from the last term of the AP: $10, 7, 4, . . . , -62$ .
- Method 1: Find total terms ( $n=25$ ), then calculate the 15th term from the start.
- Method 2: Reverse the AP: $-62, -59, -56, . . .$
- $a = -62, d = 3, n = 11$
- $a_{11} = -62 + (11 - 1) \times 3 = -62 + 30 = -32$

Gauss's Method: To sum numbers from $1$ to $100$ , Gauss wrote the sum twice (once in reverse) and added them to find constant sums of $101$ . $S = 1 + 2 + . . . + 99 + 100$ $S = 100 + 99 + . . . + 2 + 1$ $2S = 100 \times 101 \rightarrow S = 5050$
General Sum Formulas:
- Standard formula: $S = \frac{n}{2} [2a + (n - 1)d]$
- Using last term ( $a_n$ ): $S = \frac{n}{2} (a + a_n)$
- Using terminal term ( $l$ ): $S = \frac{n}{2} (a + l)$
Relationship between $a_n$ and $S_n$ : $a_n = S_n - S_{n-1}$

Example 11: Sum of first $22$ terms of $8, 3, -2, . . .$
- $a = 8, d = -5, n = 22$
- $S_{22} = \frac{22}{2} [2(8) + (22 - 1)(-5)] = 11 [16 - 105] = 11(-89) = -979$
Example 13: How many terms of $24, 21, 18, . . .$ must be taken for a sum of $78$ ?
- $78 = \frac{n}{2} [48 + (n - 1)(-3)] \rightarrow 156 = n(51 - 3n) \rightarrow 3n^2 - 51n + 156 = 0$
- $n^2 - 17n + 52 = 0 \rightarrow (n - 4)(n - 13) = 0$
- Two answers: $n = 4$ or $n = 13$ . This happens because terms from the 5th to the 13th cancel each other out to zero.
Sum of first n positive integers: $S_n = \frac{n(n + 1)}{2}$

Divisibility: To find the sum of the first $40$ positive integers divisible by $6$ , use $a=6, d=6, n=40$ .
Compound Interest vs. Simple Interest: Simple interest sequences ( $80, 160, 240, . . .$ ) form an AP, whereas compound interest sequences do not.
Coordinate/Geometry Applications:
- Spiral Length: A spiral made of 13 semicircles with increasing radii ( $0.5, 1.0, 1.5, . . .$ ). Total length is the sum of circumferences $l = \pi(r_1 + r_2 + . . . + r_{13})$ .
- Log Stacking: $20$ logs in the bottom row, $19$ in next, etc. Total logs $200$ . Solve for $n$ in $200 = \frac{n}{2} [2(20) + (n - 1)(-1)]$ .
Potato Race: Distance for first potato is $2 \times 5$ , second is $2 \times (5 + 3)$ . The sequence is $10, 16, 22, . . .$ for $10$ terms.

Definition: Each term is obtained by adding $d$ to the previous one.
General Term: $a_n = a + (n - 1)d$
Sum of n Terms: $S = \frac{n}{2} [2a + (n - 1)d]$ or $S = \frac{n}{2} (a + l)$
Arithmetic Mean: If $a, b, c$ are in AP, then $b = \frac{a + c}{2}$ , where $b$ is the arithmetic mean of $a$ and $c$ .