Geometric Progressions

Transcript

Geometric Progressions A sequence where ratio of two consecutive terms is constant. Here: a is the first term, and r is the ratio between the terms (called the "common ratio") For GP 1, 2, 4, 8, 16, 32, 64, 128, …. Here, First term = a = 1 Common ratio = r = (2^𝑛𝑑 𝑡𝑒𝑟𝑚)/(1^𝑠𝑡 𝑡𝑒𝑟𝑚) = 2/1 = 2 Note: r can be Positive, negative, or a fraction But r is never 0 Let’s look at an example of a book Growing Pattern of Squares Counting number of squares, we get 3, 6, 12, 24, …. We notice that the number of squares are multiplied by 2 each step Can you predict the number of squares in Stages 5 and 6 of the pattern? In Stages 10, 11 and 12? In Stage 20? At any stage? In this pattern, the squares go: 3, 6, 12, 24, … Here, First term = a = 3 Common ratio = r = 6/3 = 2 Now, 5th term = 4th term × Common ratio = 24 × 2 = 48 6th term = 5th term × Common ratio = 48 × 2 = 96 But finding thr 10th, 11th, 12th, 20th term is complicated We need a formula for that Let’s look at that