Arithmetic Progressions • Topic 3 of 5

General Form and nth Term of an AP

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

Solved Examples

Worked Example

Find the 15th term of the AP: 7, 13, 19, 25, …

Solution

First term $ a = 7 $, common difference $ d = 13-7 = 6 $
$ a_n = a + (n-1)d $
$ a_{15} = 7 + (15-1)×6 = 7 + 14×6 = 7 + 84 = 91 $

Answer: The 15th term is 91.

Worked Example

Which term of the AP: 100, 97, 94, 91, … is 76?

Solution

$ a = 100 $, $ d = 97-100 = -3 $
$ a_n = 100 + (n-1)(-3) = 76 $
$ 100 - 3(n-1) = 76 $
$ -3(n-1) = 76 - 100 = -24 $
$ n-1 = \frac{-24}{-3} = 8 $
$ n = 9 $

Answer: 76 is the 9th term.

Worked Example

The 8th term of an AP is 37 and the 12th term is 57. Find the AP.

Solution

$ a_8 = a + 7d = 37 $ …(1)
$ a_{12} = a + 11d = 57 $ …(2)
Subtract (1) from (2): $ 4d = 20 $ → $ d = 5 $
From (1): $ a + 35 = 37 $ → $ a = 2 $
AP: 2, 7, 12, 17, 22, …

Answer: AP is 2, 7, 12, 17, 22, …

Key Points

General form: $ a, a+d, a+2d, a+3d, \dots $
nth term formula: $ a_n = a + (n-1)d $
To find a missing term, use arithmetic mean: middle term = (sum of neighbors)/2
To check if a number belongs to an AP, solve $ N = a + (n-1)d $; if $ n $ is a positive integer, it belongs.
You can form an AP from two given terms by finding $ a $ and $ d $.

Quick Check — 4 MCQs

Tap an option to check your answer0 / 4

Q1.The $n$th term of an AP is:

Explanation: $a_n=a+(n-1)d$.

Q2.The 10th term of $2,5,8,\dots$ is:

Explanation: $2+9\cdot3=29$.

Q3.Which term of $3,7,11,\dots$ is $35$?

Explanation: $3+(n-1)4=35\Rightarrow n=9$.

Q4.The $n$th term of $1,3,5,\dots$ is:

Explanation: Odd numbers.

Practice more MCQs → 📝 Assignment · 35 marks