Sunday, July 14, 2013

What is so special about Arithmetic Progression?

2:27 AM
An arithmetic progression is a series of terms such that the difference between any two consecutive terms is always constant. For example numbers 2, 5, 8, 11,...etc. form an arithmetic progression because every subsequent term is 3 more than the previous term. The constant difference between any two terms in an arithmetic progression is known as the “Common Difference” of the arithmetic progression.
1. Mathematically the common difference can be represented as the difference between any two consecutive terms i.e. $\usepackage{color} \definecolor{Myblue}{rgb}{0.27,0.38,0.5} \color{Myblue} d= t_{n+1} - t_n$
2. The first term of an arithmetic progression is denoted by $\usepackage{color} \definecolor{Myblue}{rgb}{0.27,0.38,0.5} \color{Myblue} a$ and all subsequent terms by $\usepackage{color} \definecolor{Myblue}{rgb}{0.27,0.38,0.5} \color{Myblue} a+d, a+2d, ...$etc. where $\usepackage{color} \definecolor{Myblue}{rgb}{0.27,0.38,0.5} \color{Myblue} d$ is the common difference of the arithmetic progression
3. The formula for the $\usepackage{color} \definecolor{Myblue}{rgb}{0.27,0.38,0.5} \color{Myblue} n^{th}$ term of an arithmetic progression is:
$\usepackage{color} \definecolor{Myblue}{rgb}{0.27,0.38,0.5} \color{Myblue} a_n=a+(n-1)d$
1. The formula for the sum of first  $\usepackage{color} \definecolor{Myblue}{rgb}{0.27,0.38,0.5} \color{Myblue} n$ term of an arithmetic progression is:
$\usepackage{color} \definecolor{Myblue}{rgb}{0.27,0.38,0.5} \color{Myblue} S_n = n(a+ \frac{(n-1)d}{2})$
• If three numbers $\usepackage{color} \definecolor{Myblue}{rgb}{0.27,0.38,0.5} \color{Myblue} a_1, a_2, a_3$ are in Arithmetic Progression, the middle number is the average of the first and the third term.
$\usepackage{color} \definecolor{Myblue}{rgb}{0.27,0.38,0.5} \color{Myblue} a_2=\frac{a_1+a_3}{2}$

Example:
5, 7, 9, 11, 13 ...
What is the 11 th term of the Arithmetic Progression shown above?
A) 15
B) 20
C) 25
D) 29
E) 30
Solution:

Here we see that each term is two more than its previous term.
So, this is an arithmetic progression.
Here, $\usepackage{color} \definecolor{Myblue}{rgb}{0.27,0.38,0.5} \color{Myblue} d=2, a=5$
Hence, for $\usepackage{color} \definecolor{Myblue}{rgb}{0.27,0.38,0.5} \color{Myblue} n=11,$
we’ll have, $\usepackage{color} \definecolor{Myblue}{rgb}{0.27,0.38,0.5} \color{Myblue} t_{11} = 5 + (11-1) \times 2 = 25$