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Three Numbers in Geometric Progression - Maths

by Mohan Kumar
(Rajaji Nagar)










































three numbers in geometric progression, their sum is 7 and the sum their reciprocals is 7/5 find the terms. Search instead for three numbers in geometric progression, thier sum is 7 and the sum thier reciprocals is 7/5 find the terms

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Apr 05, 2012
Three Numbers in Geometric Progression
by: Staff


Part I

Question:
by Mohan Kumar
(Rajaji Nagar)


three numbers in geometric progression, their sum is 7 and the sum their reciprocals is 7/5 find the terms. Search instead for three numbers in geometric progression, their sum is 7 and the sum their reciprocals is 7/5 find the terms



Answer:


A geometric sequence has the (general) form:

a_n = a_1 * (r)^(n - 1)


a_n = a with a subscript of n (this is the nth term in the sequence)
a_1 = a with a subscript of 1 (this is the 1st term in the sequence)

n = number of terms

r = the common ratio

The three numbers are:

1st number: a_1

2nd number: a_2

3rd number: a_3


a_1 = a_1

a_2 = a_1 * (r)^(2 - 1) = a_1 * (r)^(1) = a_1 * (r)

a_3 = a_1 * (r)^(3 - 1) = a_1 * (r)^(2)

Since the sum of the three numbers is 7

a_1 + a_2 + a_3 = 7

a_1 + a_1 * (r) + a_1 * (r)^(2) = 7

a_1(1 + r + r^2) = 7



Since the sum of the three reciprocals is:


(1/a_1) + (1/a_2) + (1/a_3) = 7/5

(1/a_1) + (1/[a_1 * (r)] + (1/ [a_1 * (r)^(2)] = 7/5

(1 + r + r²) / (a_1 * r²) = 7/5


There are two equations with two unknowns.

The next step is to solve the equations for the two unknowns.



Divide the 1st equation (sum of the three numbers = 7) by the 2nd equation (sum of the three reciprocals = 7/5)

[a_1(1 + r + r^2) = 7] ÷ [(1 + r + r²) / (a_1 * r²) = 7/5]

[a_1(1 + r + r^2)] / [(1 + r + r²) / (a_1 * r²)] = 7 / (7/5)

[a_1 * (1 + r + r²) * (a_1 * r²)] / (1 + r + r²) = (7 * 5) / 7

[a_1 * (a_1 * r²) * (1 + r + r²) ] / (1 + r + r²) = (7 * 5) / 7

[a_1 * (a_1 * r²)] * [(1 + r + r²) / (1 + r + r²)] = 5 * (7 / 7)

[a_1 * (a_1 * r²)] * (1) = 5 * (1)

[a_1 * (a_1 * r²)] = 5

(a_1)² * r² = 5

(a_1 * r)² = 5

√(a_1 * r)² = √5

a_1 * r = √5

a_1 * r = ±√5

a_1 = (±√5)/r


Substitute value for a_1 value into the 1st equation

a_1(1 + r + r^2) = 7

[(±√5)/r ] * (1 + r + r^2) = 7


Solve for r

(±√5) * [(1 + r + r²) /r ] = 7

(±√5) * [(1 + r + r²) /r ] = 7

(±√5) * (1 + r + r²) = 7*r

±√5 + (±√5)r + (±√5)r² = 7r

±√5 + (±√5)r + (±√5)r² - 7r = 0

(±√5)r² + (±√5)r - 7r ±√5 = 0

(±√5)r² + [±√(5) - 7]r ±√5 = 0



This is a quadratic equation

----------------------------------------------------------------------

Apr 05, 2012
Three Numbers in Geometric Progression
by: Staff


----------------------------------------------------------------------

Part II


Using the quadratic formula to solve it is probably the fastest way:

ax² + bx + c = 0

x = [-b ± √(b² - 4ac)]/(2a)


(±√5)r² + [±√(5) - 7]r ±√5 = 0


a = (±√5)

b = [±√(5) - 7]

c = ±√5

to make the calculations a little easier, I am going to evaluate √5

±√5 = ±2.2360679774998 = ±2.24

a = 2.24

b = -4.76

c = 2.24


r ∈ {0.70346, 1.4215}



if r = 0.70346

a_1 = √(5)/r


a_1 = 2.24/0.70346 = 3.18426


a_2 = a_1 * (r)

a_2 = 3.18426 * (0.70346) = 2.2399995396


a_3 = a_1 * r²

a_3 = 3.18426 * (0.70346)² = 1.575750076127



The sum of the three terms = 7.0

a_1 + a_2 + a_3 = 3.18426 + 2.2399995396+ 1.575750076127 = 7.0



the sum of the reciprocals of all three terms = 7/5

(1/a_1) + (1/a_2) + (1/a_3) = 1/3.18426 + 1/2.2399995396 + 1/1.575750076127 = 1.4 = 7/5

------------------------------------------------------------------


if r = 1.4215

a_1 = √(5)/r


a_1 = 2.24/1.4215 = 1.5758002110447


a_2 = a_1 * (r)

a_2 = 1.5758002110447* (1.4215) = 2.2399995396


a_3 = a_1 * r²

a_3 = 1.5758002110447 * (1.4215)² = 3.1841600000001


The sum of the three terms = 7.0

a_1 + a_2 + a_3 = 1.5758002110447 + 2.2399995396 + 3.1841600000001 = 7.0



the sum of the reciprocals of all three terms = 7/5

(1/a_1) + (1/a_2) + (1/a_3) = 1/1.5758002110447 + 1/2.2399995396 + 1/3.1841600000001 = 1.4 = 7/5


>>> The final answer is:

if r = 0.70346

terms = 3.18426, 2.2399995396, 1.575750076127


if r = 1.4215

terms = 1.5758002110447, 2.2399995396, 3.1841600000001






Thanks for writing.

Staff
www.solving-math-problems.com



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