Arithmetic Progressions

General Form of an AP: Any AP can be written as: \[ a,\ a+d,\ a+2d,\ a+3d,\ a+4d,\ \dots \] where \( a \) is the first term and \( d \) is the common difference.

nth Term Formula: The term at position \( n \) (called \( a_n \)) is given by: \[ a_n = a + (n-1)d \] This formula allows us to find any term without writing all previous terms.

Finding Missing Terms: If some terms of an AP are missing, we can use the fact that the middle term of three consecutive terms is the arithmetic mean: For three terms \( x, y, z \) in AP: \( y = \frac{x+z}{2} \)

Checking if a number belongs to an AP: To check if a number \( N \) is a term of an AP with first term \( a \) and common difference \( d \):

• Set \( N = a + (n-1)d \)
• Solve for \( n \). If \( n \) is a positive integer, \( N \) is a term of the AP.

GENERAL FORM OF AP
   Position:  1st    2nd    3rd    4th    5th    ...    nth
   Term:      a   →  a+d  →  a+2d → a+3d → a+4d  ...  a+(n-1)d
              +d      +d      +d      +d

FINDING THE nth TERM (Formula in Action)
   Example: AP = 4, 9, 14, 19, 24, ...
   a = 4, d = 5
   
   a₁ = 4 + (1-1)×5 = 4
   a₂ = 4 + (2-1)×5 = 9
   a₃ = 4 + (3-1)×5 = 14
   a₁₀ = 4 + (10-1)×5 = 4 + 45 = 49

CHECKING IF A NUMBER BELONGS TO AN AP
   Is 78 a term of AP: 6, 13, 20, 27, ...?
   Step 1: a = 6, d = 7
   Step 2: 78 = 6 + (n-1)×7
   Step 3: 78 - 6 = (n-1)×7 → 72 = (n-1)×7 → n-1 = 72/7 = 10.285...
   Step 4: n is NOT an integer → 78 is NOT a term

ARITHMETIC MEAN (Middle Term)
   Three consecutive terms:  ____ , 12 , 20
   Middle term = (first + third)/2
   12 = (first + 20)/2 → 24 = first + 20 → first = 4
   AP: 4, 12, 20

What is Summation? The sum of the first \( n \) terms of an AP is denoted by \( S_n \). It is the total when you add all terms from the first to the nth term.

Formula for Sum of First n Terms: There are two common formulas: \[ S_n = \frac{n}{2} [2a + (n-1)d] \] \[ S_n = \frac{n}{2} (a + l) \] where \( l = a_n = a + (n-1)d \) is the last term.

Which formula to use?

• Use \( S_n = \frac{n}{2}[2a + (n-1)d] \) when you know \( a, d, n \).
• Use \( S_n = \frac{n}{2}(a + l) \) when you know \( a, l, n \).

Sum of Finite AP: For a finite AP with \( n \) terms, the sum is always a finite number.

Real-life applications:

• Total savings after \( n \) months with fixed monthly increase
• Total number of seats in an auditorium with increasing rows
• Distance traveled in \( n \) seconds with constant acceleration
• Total production over several years with fixed annual increase

SUM OF FIRST n TERMS (Visual Representation)
   AP: 2, 5, 8, 11, 14 (a=2, d=3, n=5)
   
   Write terms forward:   2  +  5  +  8  + 11  + 14 = 40
   Write terms backward: 14 + 11  +  8  +  5  +  2 = 40
   Add column by column: 16 + 16  + 16  + 16  + 16 = 80
   
   Each pair sum = a + l = 2 + 14 = 16
   Number of pairs = n = 5
   Total of both rows = n × (a + l) = 5 × 16 = 80
   So S_n = 80 ÷ 2 = 40 ✓

APPLICATION: SEATING ARRANGEMENT
   Row 1: 10 seats
   Row 2: 13 seats
   Row 3: 16 seats
   Total seats in 8 rows:
   a=10, d=3, n=8
   S₈ = 8/2[2×10 + (8-1)×3]
      = 4[20 + 21] = 4×41 = 164 seats

SUM FORMULAE QUICK REFERENCE
   ┌────────────────────────────────────────┐
   │ S_n = n/2 [2a + (n-1)d]                │
   │ S_n = n/2 (a + l) where l = last term  │
   └────────────────────────────────────────┘

Special Results (Sums of Special Sequences):

1. Sum of first n natural numbers: \( 1 + 2 + 3 + \dots + n = \frac{n(n+1)}{2} \) This is an AP with \( a = 1, d = 1 \).

2. Sum of first n even numbers: \( 2 + 4 + 6 + \dots + 2n = n(n+1) \) This is an AP with \( a = 2, d = 2 \).

3. Sum of first n odd numbers: \( 1 + 3 + 5 + \dots + (2n-1) = n^2 \) This is an AP with \( a = 1, d = 2 \).

Word Problem Strategies:

• Read the problem carefully and identify what forms the AP.
• Find the first term \( a \) and common difference \( d \).
• Determine whether the problem asks for a particular term (\( a_n \)) or a sum (\( S_n \)).
• Translate words like "each year increase by fixed amount" → \( d \)
• "after n years" → look for \( a_n \) or \( S_n \)
• "total after n days" → usually \( S_n \)

Consecutive integer problems: Three consecutive terms in AP can be taken as: \( a-d, a, a+d \) Four consecutive terms: \( a-3d, a-d, a+d, a+3d \) This simplifies algebra.

SPECIAL SUM FORMULAE (Visual Proof)
   
   SUM OF FIRST n NATURAL NUMBERS
   1 + 2 + 3 + ... + n = n(n+1)/2
   
   Example: n=4 → 1+2+3+4 = 10
   Formula: 4×5/2 = 10 ✓
   
   SUM OF FIRST n EVEN NUMBERS
   2 + 4 + 6 + ... + 2n = n(n+1)
   
   Example: n=4 → 2+4+6+8 = 20
   Formula: 4×5 = 20 ✓
   
   SUM OF FIRST n ODD NUMBERS
   1 + 3 + 5 + ... + (2n-1) = n²
   
   Example: n=4 → 1+3+5+7 = 16
   Formula: 4² = 16 ✓

STRATEGY FOR WORD PROBLEMS
   
   ┌─────────────────────────────────────┐
   │ PROBLEM → Identify a and d          │
   │         ↓                           │
   │   Find term?  Use a_n = a+(n-1)d    │
   │   Find sum?   Use S_n formula       │
   │   Find n?     Solve quadratic       │
   └─────────────────────────────────────┘

TAKING CONSECUTIVE TERMS IN AP
   
   3 terms:  a-d,  a,  a+d   (sum = 3a)
   4 terms:  a-3d, a-d, a+d, a+3d (sum = 4a)
   5 terms:  a-2d, a-d, a, a+d, a+2d (sum = 5a)