Geometric progressions

Basic concepts of Geometric Progressions and exercise

1. Geometic Progressions

2. Geometric Progression If in a sequence the ratio of any term to its preceding term is constant, it is called geometric progression. The constant ratio is called the comman ratio. Eg . 5, 15, 45, 135, 405, … is a GP Whose common ratio is 3

3. Geometric Progression If a is the first term and r is the common ratio o a GP, it can be expressed as: a, ar , ar 2 , ar 3 , ar 4 , … this general form of GP

4. n th term of GP Let a, ar , ar 2 , ar 3 , ar 4 , … be a GP Here T 1 = a = ar 1-1 T 2 = ar = ar 2-1 T 3 = ar 2 = ar 3-1 T 4 = ar 3 = ar 4-1 T n = a r n-1

5. Sum of n terms of GP Let a, ar , ar 2 , ar 3 , ar 4 , … be a GP S n = a + ar + ar 2 + ar 3 + … + a r n-1 Sn = a (1 – r n ) / (1 – r) {if r < 1 Sn = a ( r n – 1) / (r – 1) {if r > 1

6. Ex.1 Find the required term of the following progressions: ( i ) 2, 6, 18, 54, … (9 th term) (ii) 1024, -512, 256, -128, … (10 th term) (iii) 5, 25, 125, 625, … (40 th term)

7. Ex.2 Find the sum up to the required term of the following progressions: ( i ) 1, 2, 4, 8, 16, … (up to 12 terms) (ii) 2, 10, 50, 250 , … (up to 9 terms) (iii)8, 4, 2, 1, … (40 th term)

8. Ex.3 The fourth term of a GP is 4 and the product of second and the fourth terms is 1. Find the sixth term and sum to first 6 terms.

9. Ex.4 The fourth and the seventh terms of a GP are 72 and 576. Find the sum of first n terms.

10. Ex.5 Three numbers are in GP. Their sum and product are 28 and 512 respectively. Find the numbers.