Sequences & Progressions

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 Formula for $\bold{n^{\text{th}}}$ 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 $\dfrac{1}{3}$. $27 \times \left( \dfrac{1}{3} \right)^{(n - 1)}$ 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 $

$\therefore x_{1} = a$
$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
$3, 7, 11, ...$ $a = 3, d = 4$ $3 + (n - 1)d$
$-89, -72, -55, ...$ $a = -89, d = 17$ $89 + (n - 1) \times 17$
$100, 96, 92, ...$ $a = 100, d = -4$ $100 + (n - 1) \times (-4)$


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 $= S_{n} = n \times \text{Average} = n \times \left( \dfrac{x_{1} + x_{n}}{2} \right) = \dfrac{n}{2} \times [2a + (n - 1)d]$

$4$) 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).

$x_{1} + x_{n} = a + a + ( n - 1)d = 2a + (n - 1)d$
$x_{2} + x_{n - 1} = a + d + a + (n - 2)d = 2a + (n - 1)d$
$x_{3} + x_{n - 2} = a + 2d + a + (n - 3)d = 2a + (n - 1)d$

Example $1$

How many terms are in the AP $3, 9, 15, 21, ....., 225$ ?

Solution

$d = 9 - 3 = 6$

$ n = \dfrac{x_{n} - x_{1}}{d} + 1 = \dfrac{225 - 3}{6} + 1 = 38$

Answer: $38$ terms