Arithmetic Mean
Given two numbers a and b . We can insert a number A between them so that a , A , b is an A . P . Such a number A is called the arithmetic mean ( A . M .) of the numbers a and b .
A . M . between two numbers a and b is their average or A = ( a + b )/ 2 In the case above , n = 10 , a = 1000 , d = 100 , So , S = 10 / 2 [ 2 * 1000 + ( 10-1 )* 100 ] = 5 *[ 2000 + 900 ] = 14500 .
Geometric Progression
Let ’ s suppose a student is asked to double the number of maths question practice every month . First month he practiced 50 questions . Number of questions he practiced over next few months will be 50 , 100 , 200 , 400 , 800 …
There is a pattern in this sequence ; here if you divide any two consecutive numbers you will get same value . 100 / 50 = 200 / 100 = 2 & thus this sequence is also Progression . This type of progression is called Geometric Progression
A sequence a1 , a2 , a3 , …, an , … is called geometric progression , if each term is non-zero and ak + 1 / ak = r , for k ≥ 1 and r is constant . Also GP can be written as : a , ar , ar 2 , ar 3 ,…., where a is called the first term and r is called common ratio of the G . P .
Numerical : Tell if the sequence is in GP or not ? 1,3,9,27,51 Solution : A sequence a1 , a2 , a3 , …, an , … is called geometric progression , if each term is non-zero and ak + 1 / ak = r , for k ≥ 1 and r is constant 3 / 1 = 9 / 3 = 27 / 9 ≠ 51 / 27
Since all the ratios are not equal , it is not a GP .