1. Introduction

Sequence is a set of numbers that follow a logical rule or pattern. This logical rule/pattern need not necessarily translate into a formula.

Progression is a set of numbers that follow a logical rule, which has a formula to calculate the $ n^{\text{th}} $ term. This lesson will cover the different forms of progressions that we are tested on with special focus on the $3$ basic forms of progressions – Arithmetic Progression (AP), Geometric Progression (GP), and Harmonic Progression (HP).

Terms	Pattern	nth term	Sequence?	Progression?
$5$, $8$, $11$, $14$, $17$ , ...	Difference between consecutive terms is $3$.	$5 + 3 \times (n - 1) $	Yes	Yes
$27, 9, 3, 1,$ $ \dfrac{1}{3} $, $ \dfrac{1}{9} $, ...	Every consecutive term is a multiple of ($\dfrac{1}{3}$) .	$27 \times$ ( $\dfrac{1}{3} )^{(n - 1)}$	Yes	Yes
$ 2, 3, 5, 7, 11, 13, 17$, ...	List of all prime numbers.	No direct formula	Yes	No
$1, 1, 2, 3, 5, 8, 13, ..$.	Sum of previous $2$ numbers.	No direct formula	Yes	No

2. Arithmetic Progression

If the difference between every two consecutive terms of a sequence is constant, then the sequence is said to be in Arithmetic Progression (AP).

If $ x_1, x_2, x_3, x_4, ..., x_n $ are in Arithmetic Progression, then $ x_2 - x_1 = x_3 - x_2 = ... = x_n - x_{n - 1} $

Where $a$ is the first term and $d$ is the common difference, the terms of the AP will be

$a$, $a + d$, $a + 2d$, $a + 3d$, ... , $a$ + ($n$ - 1)$d$

∴ $x_1 = a$
$x_2 = a + d$
$x_3 = a + 2d$
$x_4 = a + 3d$
$x_n $ $=$ $a$ + ($n$ – 1)$d$

For the following examples of AP, not the first term $ (a) $ and common difference $ (d) $.

Arithmetic Progression	*a & d*	n$^{\text{th}}$ term
$3$, $7$, $11$, ...	$a = 3$, $d = 4$	$3+(n-1) \times 4$
$-89$, $-72$, $-55$, ...	$a$ $= -89$, $d = 17$	$-89+(n-1) \times 17$
$100$, $96$, $92$, ...	$a = 100$, $d$ $= -4$	$100+(n-1) \times (-4)$

2.1 Terms & Formulae for AP

Where $a$ and $d$ are the first term and common difference respectively in an AP with $n$ terms,

1) $ n^{\text{th}} $ $=$ $a$ + ($n$ – 1)$d$

2) Average of an AP $=$ Average of First and Last terms $=$ $\dfrac {x_1+x_n}{2}$

3) Sum of terms of an AP $=$ $ \bold{S_n} $ $=$ $n \times \text{Average} = n \times \left( \dfrac{x_1 + x_n}{2} \right) = \dfrac {n}{2} \times [ 2a + (n-1)d ]$

4) Number of terms in an AP $=$ $ n $ $=$ $\dfrac {\text{Last Term - First Term}}{\text{Common Difference}}+1 = \dfrac {x_n- x_1}{d}+1$

2.2 Properties of AP

1) If each term of an AP is added, subtracted, multiplied or divided by a constant, then the resulting sequence is also in AP.

2) In two APs with the same number of terms, if the corresponding terms by position in the two APs are added/subtracted, the resulting sequence will also be in AP.

3) Average of an AP is its median.

4) If $ a $, $b$ and $c $ are in AP, then $ b = \dfrac{a+c}{2}$

5) Sum of the first and last terms equals the second and second last terms, which then equals the third and third from last terms and so on (as shown below).

$x_1 + x_n$ $=$ $a$ + $a$ + ($n$ – 1)$d$ $=$ $2a$ + ($n$ – 1)$d$
$x_2 + x_{n-1}$ $=$ $a$ + $d $+ $a$ + ($n$ – 2)$d $ $=$ $2a$ + ($n$ – 1)$d$
$x_3 + x_{n-2} $ $=$ $a$ + $2d$ + $a$ + ($n$ – 3)$d$ $=$ $2a$ + ($n$ – 1)$d$