Question

Which of these two sequences form a geometric sequence: 2x, 4x-2, 6x-4, ..... OR x+1, 2x+2, 5x+1, .....? Find the value/s of x and what is the sequenced formed?

My answer is neither is a GP since they don't have a common ratio. If this is the case, is there still any way to still get the x? Cause that is what is in the question, find the value of x and the sequence formed. Thanks.

Answer 1

In a geometric sequence, consecutive terms have a common ratio.

This can be expressed as:

a(n) / a(n-1) = q

In this case:

a2 / a1 = q

and

a3 / a2 = q


Of course q = q


a2 / a1 = a3 / a2


Sequence:

2 x , 4 x - 2 , 6 x - 4


a1 = 2 x , a2 = 4 x - 2 , a3 = 6x-4

a2 / a1 = a3 / a2

( 4 x - 2 ) / 2 x = ( 6 x - 4 ) / ( 4 x - 2 ) Multiply both sides by 2 x

4 x - 2 = 2 x ( 6 x - 4 ) / ( 4 x - 2 ) Multiply both sides by ( 4 x - 2 )

( 4 x - 2 ) ( 4 x - 2 ) = 2 x ( 6 x - 4 )

( 4 x - 2 ) ^ 2 = 2 x ( 6 x - 4 )

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Remark:

( a - b ) ^ 2 = a ^ 2 - 2 a * b + b ^ 2

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( 4 x ) ^ 2 - 2 * 4 x * 2 + 2 ^ 2 = 2 x * 6 x - 2 x * 4

16 x ^ 2 - 16 x + 4 = 12 x ^ 2 - 8 x Subtact 12 x ^ 2 to both sides

16 x ^ 2 - 16 x + 4 - 12 x ^ 2 = 12 x ^ 2 - 8 x - 12 x ^ 2

4 x ^ 2 - 16 x + 4 = - 8 x Add 8 x to both sides

4 x ^ 2 - 16 x + 4 + 8 x = - 8 x + 8 x

4 x ^ 2 - 8 x + 4 = 0

4 ( x ^ 2 - 2 x + 1 ) = 0 Divide both sides by 4

x ^ 2 - 2 x + 1 = 0

( x - 1 ) ^ 2 = 0 Take square rot to both sides

x - 1 = 0 Add 1 to both sides

x - 1 + 1 = 0 + 1

x = 1

So:

a1 = 2 x

a1 = 2 * 1 = 2


a2 = 4 x - 2

a2 = 4 * 1 - 2 = 4 - 2 = 2


a3 = 6 x - 4

a3 = 6 * 1 - 4 = 6 - 4 = 2


Tre members of a sequence are equal.

That is not a geometric sequence.


Sequence:

x + 1 , 2 x + 2 , 5 x + 1


a1 = x + 1 , a2 = 2 x + 2 , a3 = 5 x + 1


a2 / a1 = a3 / a2

( 2 x + 2 ) / ( x + 1 ) = ( 5 x + 1 ) / ( 2 x + 2 )

2 ( x + 1 ) / ( x + 1 ) = ( 5 x + 1 ) / ( 2 x + 2 )

2 = ( 5 x + 1 ) / ( 2 x + 2 ) Multiply both sides by ( 2 x + 2 )

2 * ( 2 x + 2 ) = 5 x + 1

4 x + 4 = 5 x + 1 Subtract 4 x to both sides

4 x + 4 - 4 x = 5 x + 1 - 4 x

4 = x + 1 Subtract 1 to both sides

4 - 1 = x + 1 - 1

3 = x

x = 3


So:

a1 = x + 1

a1 = 3 + 1 = 4


a2 = 2 x + 2

a2 = 2 * 3 + 2 = 6 + 2 = 8


a3 = 5 x + 1

a3 = 5 * 3 + 1 = 15 + 1 = 16


q = a2 / a1 = 8 / 4 = 2

OR

q = a3 / a2 = 16 / 8 = 2


That is a geometric sequence.

First term :

a1 = 4

Common ratio:

q = 2

Answer 2

Thank you very much. I got it now.