Mathematical Progressions

May 06, 2019 2 minutes

A mathematical progression is a sequence of numbers such that all the members of the sequence are governed by certain rule, which defines the relation between consecutive terms. There are three types of mathematical progressions:

1. Arithmetic Progression

An arithmetic progression (AP) or arithmetic sequence is a sequence of numbers such that the difference between the consecutive terms is constant.

For an AP with common difference as d and first term as $a_1$, the terms of the progression can be represented as: $a_1, a_1+d, a_1+2d, a_1+3d … a_1+(n-1)d$

Consider the following sequence for example:

  • {1,2,3,4,5}: AP with common difference as 1
  • {2,4,6,8,10}: AP with common difference as 2
  • {5,10,15,20,25}: AP with common difference as 5

For a sequence of terms ${a_1, a_2, … . a_n}$ with common difference as d

  • The $n^{th}$ term is given by : $a_n = a_1 + (n-1)d$
  • Sum of first n terms of the sequence $S_n = \frac{n}{2}(a_1+a_n)$
  • The sum of first n terms can also be represented as $S_n = \frac{n}{2}\bigl(2a_1 + (n-1)\bigr)d$
  • $N^{th}$ term can be represented as $Sum(n) - Sum(n-1)$
  • The middle term of three consecutive terms of an AP is the mean of the first and last. $a_2 = \frac{a_1 + a_3}{2}$

2. Geometric Progression

A geometric progression (GP) or geometric sequence is a sequence of numbers such that the ratio between the consecutive terms is constant.

On similar lines as mentioned above for AP, consider the first term to be a and common ratio to be r, the terms of GP can be represented as: $a, ar, ar^2, ar^3 … ar^{n-1}$

Consider the following sequence for example:

  • {2,4,8,16,32}: GP with first term as 2 and common ratio as 2
  • {5,10,20,40,80}: GP with first term as 5 and common ratio as 2

Some useful tips on GP:

  • The $n^{th}$ term is given by : $a_n = ar^{n-1}$
  • Sum of first n terms $S_n = \frac{ar^n - 1}{r-1}; r \ne 1$

3. Harmonics Progression

Harmonic Progression or harmonic series is a sequence of numbers that originally form an arithmetic progression. Consider the terms $a, a+d, a+2d … a+(n-1)d$ of an arithmetic progression with first term as a and common difference as d, the terms of the HP will be given by: $$\frac{1}{a}, \frac{1}{a+d}, \frac{1}{a+2d}… \frac{1}{a+(n-1)d}$$

Consider the following examples:

  • $1,\frac{1}{2}, \frac{1}{3},\frac{1}{4},\frac{1}{5}$; {1,2,3,4,5} forms an AP with a=1 and d=1
  • $\frac{1}{2},\frac{1}{4},\frac{1}{6},\frac{1}{8},\frac{1}{10}$; {2,4,6,8,10} forms an AP with a=2 and d=2
  • $\frac{1}{5},\frac{1}{10},\frac{1}{15}, \frac{1}{20},\frac{1}{25}$; {5,10,15,20,25} forms an AP with a=5 and d=10
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