Geometric Sequence (G.S)Definition: is called a G.S if the ratio between every two consecutive terms is constant.i . e .This constant is called common ratio ( base ) (r) rExample (1)Prove that the following sequence are G.S : solution The general form of G.S : Where : common ratio Last termThe general term : Example (2)Find solution Example (3) Find the order of the term , whose value is 3072 in the G.S solution 3072 = Example (4) Find the first term whose value less than 0.001 in the G.S ( 1, 0.2, 0.04, … … ).solution 1 Example (5) Find the number of terms of the G.S solution n = 8Example (6)A G.S it’s first term is 2 and Find G.S .solution (1) (2)Subs in (2) by (1)2 =729 G.S Example (7)G.S Find .solution Example (8)If then prove that is a G.S and Find its first three terms.solution (1) Subs in (1) Example (9)Find the third term of a G.S, in which its terms are positive and and .solution (1) (2) 4 refused Subs is (2) Example (10)A G.S whose all it’s terms are positive and Find solution a (1) (2) Subs by (2) in (1): Subs in (2) Example (11)Prove that in any G.S if , then . Solution Example (12)The sum of the first and the second terms of a G.S equals 4 and the sum of its third and fourth terms equals 36, Find the G.S.Solution (1) (2)(2) Subs in (1) r = 3 G.S(-2,6,-18, …) G.S(113,9,…… ) Example (13)A G.S in which the sum of its first three terms is 26 and , Find the G.S.Solution (1) (2)(2) Subs in (1) G.S Example (14)An increasing G.S in which and Find the G.S.Solution (1) (2)(2) r = 3 r = 2 refusedSubs in (1) G.S (2,6,18,… … )Example (15)The sum of three consecutive number in a G.S equals 21 and their product equals 64 , Find these numbers.SolutionLet the three numbers be (1) Subs in (1) Three number: 1, 4, 16 Example (16)Five positive number form a G.S where and , Find these numbers. Solution (1) (2) Subs in (2) Five number are ( 256, 128, 64, 32,16 ).Example (17)The sum of the first three positive terms of a G.S equals 28 and the sum of their multiplicative inverse is , Find these numbers.Solution (1) 16 (2)(1)(2) or Subs in (1) The three numbers are ( 4, 8, 16 ). Example (18)Find the G.S in which and also Find n.Solution (1) (2) (2) Subs in (1) G.S ( 32,16,8, … … ).The geometric mean and its relation with the arithmetic meanIF G.S b is G.M of a and c i.e Example (19)Find two positive numbers for which their sum is 30 and their geometric mean is 12.SolutionLet the two numbers be . Two numbers are ( 6 and 24 ) Example (20)Insert 6 positive G.Ms between 5 and 640. SolutionG.S where n = no of means 6 G.Ms are10 , 20 , 40 , 80 , 160 , 320 Example (21)G.S Solution2 Example (22)Let A.S Find .SolutionA.S (1) (2)Subs in (1) by (2) Example (23)A.S if their sum equals 9 and G.S , Find . SolutionA.S 3 G.S 3 4 Three numbers are Example (24)If we insert some positive geometrical means between 2 and 486 such that the sum of the last two means equals nine time the sum of the first two means, so Find the number of these means.SolutionLet the no. of means = nG.S 27 = no. of means = 4Example (25)The arithmetic mean of two number equals their geometrical mean, if the smaller number is 9 , so Find the other number.SolutionLet the two numbers be 9 , 9+x refusedExample (26)Find two positive numbers such that their arithmetic mean exceeds their geometric mean by 2.5 and the ratio between the two means is 13 : 12.SolutionLet A.M = A , G.M = GA – G = 2.5 A= G + 2.5 (1) (2)Subs in (2) by (1) 12 G + 30 =13 G 30 = G Let the two numbers be (3) (4)Subs in (4) by (3) 65 Two numbers are 20 and 45Example (27)The sum of three consecutive terms of an A.S is 15, If we subtract one from the first and one from the second and add one to the third then they become in a G.S , Find the three numbers.SolutionLet the three numbers be A.S G.S 16 = Three numbers are : Example (28)Three numbers are in a G.S such that their sum is 21, If we add to the second 1.5, then they become in an A.S Find these numbers.SolutionLet the three numbers be: G.S A.S 2 (2) Subs in (1) The three numbers are ( 3, 6, 12 ) For any two positive numberA.M Example (29)If are in G.S with positive terms, then prove that .SolutionG.S Let be two quantities their G.M = y A.M = (1)Let be two quantitiesTheir G.M = 3z A.M = (2) Example (30)If is in an A.S whose terms are positive , then prove that : .SolutionLet be two quantitiesTheir A.M = b G.M = b (1)Let be two quantitiesTheir A.M = c G.M = C = (2)+ (2) Example (31)If 2are positive numbers then prove that: .SolutionLet be two quantities Their A.M = G.M = (1)Let be two quantitiesTheir A.M = G.M = (2)+(2) The sum of a finite number of terms of a G.S12Where s3 Then G.S could be added to infinite number of terms. Example (32)Find the sum of the first six term of the G.S (3,12,48,……….).Solution Example (33)Find the sum of the terms of the G.S ( 384,192, 96,… …, ).Solution Example (34)Find the sum of a G.S, whose first term is 243, and its last term is one and the number of its terms is 6.Solution Example (35)Find the sum of the first six terms of a G.S, whose terms are all positive, the first term is 4 and Solution Example (36)How many terms must be taken from the G.S ( 0.2, -0.8, 3.2, …) starting with the first term such that its sum is 655.4 Solution Example (37)A G.S in which its first term is 2 and its fourth term is 54, Find the least number of terms of this G.S starting with its first term such that their sum is greater than 5000. Solution G.S (2,6,18,… … ) 75 Example (38)A G.S in which the sum of its fourth and sixth terms is 120 and the sum of its fifth and seventh term is 240, Find the G.S and the sum of its first ten terms. Solution (1) (2)Subs in (1) 8 G.S (3,6,12,….) Example (39)If the sum of the first five terms of a G.S is 31 and the sum of next five terms is 992, Find this G.S and the product of the first ten terms. Solution (1) (2) Subs in (1) G.S (1,2,4,… … ) Example (40)If the sum of the first n – term of a G.S is given by : , so Find this G.S and its .SolutionSn-Sn-1=Tn G.S (384,96,24,….) Example (41)Prove that : G.S could be added to infinity.Solution = Example (42)Let L be the sum of the first ten terms of a G.S. M be the sum of the first twenty terms of this G.S. K be the sum of the first thirty terms of this G.S , so Prove that L,M,K + K – L are in a G.S.Solution (1) (2) = K (3)(2)(1) (4) (5) From (4) and (5) Example (43)Find the sum of G.S ( ) to infinity.Solution Example (44)Change 0 . into a common fraction using G.S.Solution0 . Example (45)Change 3.4 into a common fraction using G.S.Solution3.4 Example (46)Find the G.S , whose sum to equal 25 and the difference between is 4. Solution (1) (2)Subs in (2) by (1) 25 Subs in (2) G.S (10, 6, ) Example (47)Find the G.S in which each of its terms equals 7 times the sum of the following terms to and its third term is .Solution G.S ( 24, 3, ). Example (48)A G.S in which the product of its first three terms is 64 and the sum of the second, the third and the fourth is 7, Prove that there are two G.S such that we can sum the terms of one of them to and Find this sum.Solution (1) (2)Subs in (2) by (1)4 Subs in (1) G.S Couldn’t be added to infinity Example (49)If the sum to of a G.S is 4 and the sum of the cubes of its terms to is 192, Find this G.S.SolutionG.S (1)G.S (2)Subs in (2) by (1) 64 2 Subs in (1) G.S Example (50)If to , R.T.P .Solution (1) (2)Subs in (2) by (1) Example (51)Find the sum of the first n – terms of G.S (10, S, ), then find the sum of this sequence to if then find the value of n. SolutionG.S (10, S, ), 20 Example (52)The sum of the odd terms of a G.S to and he sum of the even terms of the same G.S to , Find the G.S, then Prove that the difference between the sum of an infinite number of terms of this G.S and the sum of the first eight terms of this G.S equal .SolutionG.S (1) (2)Subs in (2) by (1) Subs in (1) G.S ( 8,4,2,…. ) Example (53)A tank which has 6138 liters of water it leaks 6 liters in the first day, 12 liters in the second day and 24 liters in the third day, and so on, after how many days will the tank be empty?SolutionG.S After 10 days.Example (54)An employee starts his job with a salary of L.E 3600 which increases every year by a ratio of of the preceding year, how much will his salary be after 11 year? And what is total sum paid to him during this 11 year?SolutionIncrease: =5864 Example (55)The production of an oil well is decreasing with the rate of 5% compared to the preceding year if the production of the first year’s oil is 48000 barrels , then Find the total sum of the well production in the first five years and also the maximum production of the well. SolutionDecrease: Problems on A.S and G.S Example (56)Find the order of the term of the A.S (130, 134,138, …) which is equal to the fifth term of the G.S (2,6,18,….).SolutionG.S (2,6,18,….) A.S (130, 134,138, …) =162 A.S =G.SExample (57)How many terms must be taken from the A.S (22,18,14,….) such that their sum is equal to the sum of the terms of the G.S (18,12,8,……) to SolutionG.S (18,12,8,……) A.S (22,18,14,….) Example (58)The sum of the first 15 terms of an A.S is 225 and its are in G.S Find the A.S.SolutionA.S (1), , Subs in (1) A.S (1, 3, 5, ……………. ).Example (59)A + ve G.S, if the sum of its first 4 terms is 120 and A.M. of its third and fifth terms is 5, then find the sequence and the sum of an infinite terms starting from its first term.SolutionG.S (1) (2)(1) refused G.S Example (60)A +ve G.S and its common ratio less than one and the A.M of the third and the fifth terms is 60 and the +ve G.M of them is 48 , Find the sequence , then Prove that the sum of any number of terms of this sequence does not exceed 768.Solution+ve G.S A.M=60 (1)G.M of (2) refusedSubs in (2) Example (61)Let G.S and The common ratio of the G.SThe G.S if the sum of the first six terms is 189.Solution (1)A.S (2)Subs in (1) = = 1 : 2 : 4ii) iii) Example (62)Three positive numbers are in an A.S this sum is 15. If we multiply the smallest by 2 and add 7 to the middle and 17 to the greatest then the numbers become in a G.S , Find these numbers.SolutionLet the three numbers be A.S ( 5)G.S ( ) Three numbers: 3 , 5 , 7 Example (63)The sum of the first three terms of a G.S , whose terms are integers is 35, if the first number is multiplied by 12, the second by 5 and the third by 2 , then they become in A.S, Find the G.S then find the order of the term whose value is 1280 in this sequence.SolutionG.S (1)A.S 5 Subs in (1) refusedG.S (5 , 10 , 20 , ….) Example (64)There are two sequences , one of them is G.S and the other is an A.S, let the first terms of each equals 3, and the second term of the G.S equal the fourth term of the A.S, and the third term of the G.S equal the tenth terms of the A.S, Find the two sequences.SolutionA.S ( 3,) G.S (1) = 3 (2)(2)(1) Subs in (2) A.S G.S