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.