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Geometric Sequence Help Please











































find all values of x such that x−3, x+3, and 3x−3 form a geometric sequence. Give your answers in increasing order.

x can equal or .

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Apr 19, 2011
Geometric Sequence – Solve for x
by: Staff


Part II


solve for x


the common ratio must be the same for all terms in the sequence

r_n = a_2/a_1 = a_3/a_2

a_2/a_1 = a_3/a_2

(x + 3)/( x - 3) = (3x - 3)/( x + 3)


Multiply both sides of the equation by (x + 3)

( x + 3) * (x + 3)/( x - 3) = ( x + 3) * (3x - 3)/( x + 3)

( x + 3) * (x + 3)/( x - 3) = (3x - 3) * (x + 3)/( x + 3)

( x + 3) * (x + 3)/( x - 3) = (3x - 3) * 1

( x + 3) * (x + 3)/( x - 3) = (3x - 3)


Multiply both sides of the equation by (x - 3)

(x - 3) * ( x + 3) * (x + 3)/( x - 3) = (x - 3) * (3x - 3)

( x + 3) * (x + 3) * (x - 3)/( x - 3) = (x - 3) * (3x - 3)

( x + 3) * (x + 3) * 1 = (x - 3) * (3x - 3)

( x + 3) * (x + 3) = (x - 3) * (3x - 3)


Multiply the factors on the left side of the equation

x² + 6x + 9 = (x - 3) * (3x - 3)


Multiply the factors on the right side of the equation

x² + 6x + 9 = 3x² - 12x + 9


subtract 9 from each side of the equation

x² + 6x + 9 - 9 = 3x² - 12x + 9 - 9

x² + 6x + 0 = 3x² - 12x + 0

x² + 6x = 3x² - 12x


subtract x² from each side of the equation

x² - x² + 6x = 3x² - x² - 12x

0 + 6x = 3x² - x² - 12x

6x = 3x² - x² - 12x

6x = 2x² - 12x


subtract 6x from each side of the equation

6x - 6x = 2x² - 12x - 6x

0 = 2x² - 12x - 6x

0 = 2x² - 18x

2x² - 18x = 0


Factor the left side of the equation

2x*(x - 9) = 0


Divide each side of the equation by 2

2x*(x - 9)/2 = 0/2

x*(x - 9)*(2/2) = 0/2

x*(x - 9)*(1) = 0/2

x*(x - 9) = 0/2

x*(x - 9) = 0





x can have only two possible values


First value of x

x = 0


Second value of x

(x - 9) = 0

x - 9 + 9 = 0 + 9

x + 0 = 0 + 9

x = 0 + 9

x = 9


the final answer is:

x ∈ {0, 9}



Check the work. Substitute 0 and 9 for x the equation for the common ratio

(x + 3)/( x - 3) = (3x - 3)/( x + 3)

If x = 0

(0 + 3)/( 0 - 3) = (3*0 - 3)/( 0 + 3)

3/(-3) = (- 3)/3

-1 = -1, correct


If x = 9

(9 + 3)/( 9 - 3) = (3*9 - 3)/( 9 + 3)

12/6 = (27 - 3)/( 9 + 3)

2 = 24/12

2 = 2, correct







Thanks for writing.


Staff
www.solving-math-problems.com


Apr 19, 2011
Geometric Sequence - Solve for x
by: Staff

Part I

The question:

find all values of x such that x−3, x+3, and 3x−3 form a geometric sequence. Give your answers in increasing order.

x can equal or .


The answer:

find all values of x such that x−3, x+3, and 3x−3 form a geometric sequence.



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

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r, the common ratio, can be calculated as follows:

r_n = a_n/a_n-1

(n must be greater than 1)

r_n = r with a subscript of n (this is the common ratio)

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

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


using the 1st and 2nd terms in the sequence to calculate the common ratio

a_1 = x−3

a_2 = x + 3

r_n = a_n/a_n-1

r_n = a_2/a_1

r_n = (x + 3)/( x - 3)


using the 2nd and 3rd terms in the sequence to calculate the common ratio

a_2 = x + 3

a_3 = 3x ? 3

r_n = a_n/a_n-1

r_n = a_3/a_2

r_n = (3x - 3)/( x + 3)

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