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.
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
Thank you very much. I got it now.
You are correct that neither of the given sequences form a geometric sequence because they do not have a common ratio. In a geometric sequence, each term is found by multiplying the previous term by the same number.
However, even though the sequences are not geometric, we can still find the value of x and determine the pattern formed by the sequences.
Let's analyze the first sequence: 2x, 4x-2, 6x-4, ...
To find the pattern, let's examine the differences between consecutive terms:
4x-2 - 2x = 2x-2
6x-4 - 4x-2 = 2x-2
We notice that the differences between consecutive terms are the same, 2x-2. Therefore, we can conclude that the pattern formed by the first sequence is an arithmetic sequence with a common difference of 2x-2.
Now let's analyze the second sequence: x+1, 2x+2, 5x+1, ...
To find the pattern, let's again examine the differences between consecutive terms:
2x+2 - x+1 = x+1
5x+1 - 2x+2 = 3x-1
We can see that the differences between consecutive terms are not constant, so this sequence does not follow a clear pattern.
Since the second sequence does not have a pattern, we cannot determine a specific value for x or a sequence formed by the terms.
In conclusion, the first sequence forms an arithmetic sequence with a common difference of 2x-2, while the second sequence does not form any specific pattern.