GEOMETRIC PROGRESSION (G.P) Calculation

CONTENT

• Definition of Geometric Progression
• Denotations of Geometric progression
• The nth term of a G. P.
• The sum of Geometric series
• Sum of G. P. to infinity
• Geometric mean

Definition of G. P

The sequence 5, 10, 20, 40 has a first term of 5 and the common ratio

Between the term is 2 e.g. (¹⁰/₅ or ⁴⁰/_2o = 2).

A sequence in which the terms either increase or decrease in a common ratio is called a Geometric Progression

(G. P)

G. P:     a,      ar,     ar²,        ar³ ………………

Denotations in G. P

a    = 1^st term

r    = common ratio

U_n = nth term

S_n  = sum

The nth term of a G. P

The nth term = Un

Un = ar^n-1

1^st term   = a

2^nd term = a x r =ar

3^rd term = a x r x r = ar²

4^th term = a x r x r x r =  ar³

8^thterm  =  a x r x r x r x r x r x r x r  =  ar⁷

nth term  =  a x r x r x r x ……….. ar^n-1

Example

Given the GP 5, 10, 20, 40. Find its (a) 9^th term (b) nth term

Solution

a = 5         r  = 10/5  =  2

U₉         =   ar^n-1

U₉         =   5 (2) ^9-1

            =   5 (2)⁸

            =   5 x 256   = 1,280

(b)        U_n = ar^n-1

                  = 5(2) ^n-1

GEOMETRIC PROGRESSION (G.P) CALCULATION

EVALUATION

The third term of a G.P. is 1/81.  Determine the first term if the common ratio is 1/3.

GENERAL EVALUATION /REVISION QUESTION

1. p – 6, 2p  and 8p + 20 are three consecutive terms of a GP. Determine the value of   (a) p (b) the common ratio

2. If 1  , x , 1 , y , ….are in GP , find the product of x and y

       16        4       

3.The   third  term  of  a  G.P   is  45  and  the  fifth  term  405.Find  the  G.P. if  the  common  ratio  r  is  positive.

4.Find  the  7^th  term  and  the  nth  term  of  the  progression  27,9,3,…

5.In  a  G.P, the  second  and  fourth  terms  are  0.04  and  1 respectively. Find the (a) common ratio (b) first term

WEEKEND ASSIGNMENT

1. In the 2^nd and 4^th term of a G.P are 8 and 32 respectively, what is the sum of the first four terms.  (a) 28    (b)  40    (c)  48    (d) 60

2. The sum of the first five term of the G.P. 2, 6, 18, is     (a) 484    (b) 243      (c) 242    (d) 130

3. The 4^th term of a GP is   -2/3 and its first term is 18 what is its common ratio.  (a) ½    (b) ¹/₃

(c) ^-1/₃   (d)  ^-1/₂

4. If the 2^nd and 5^th term of a G. P. are -6 and 48 respectively, find the sum of the first four terms:  (a) -45    (b) -15     (c) 15    (d) 33

5. Find the first term of the G.P. if its common ratio and sum to infinity – ³/₃ and respectively (a) 48    (b) 18    (c) 40   (d) -42

THEORY

1.The 3^rd term of a GP is 360 and the 6^th term is 1215. Find the

(i)         Common ratio         (ii) First term        (iii)        Sum of the first four terms

1b. If (3- x) + (6) + (7- 5x) is a geometric series, find two possible values for

(i)   x     (ii)  the  common  ratio,  r          (iii) the sum  of  the   G.P

2.The first term of a G. P. is 48. Find the common ratio between its terms if its sum to infinity is 36.

Reading Assignment

New General Mathematics SSS2