Determine the value of x that makes 2, 6, 5x - 2, ... a geometric sequence.

Question

Determine the value of x that makes 2, 6, 5x - 2, ... a geometric sequence.

Please answer in an organized and neat/easy to read format. And please only include the information that would be understood by a Grade 11. Thank you :)

Answer 1

The thing that makes a sequence geometric is that you multiply by the same number every time to get from one term to the next, so the ratio between any two consecutive terms has to come out the same. That single fact is really all we need here.

We already have two terms we can trust, 2 and 6, so we can find that ratio right away by dividing:

r = 6 / 2 = 3

Since the ratio has to stay 3 for the whole sequence, the third term has to be 6 multiplied by 3, which is 18. If you notice, we now have two different expressions for that same third term, 18 and 5x - 2, and if they are both the third term then they have to be equal to each other:

5x - 2 = 18

Adding 2 to both sides gives 5x = 20, and dividing both sides by 5 gives us x = 4.

It is worth checking, since it only takes a second. Putting x = 4 back in makes the third term 5(4) - 2 = 18, so the sequence is 2, 6, 18, and each term really is three times the one before it.

So, our final answer is x = 4.

Answer 2

2,6,5x-2 is a GP with r=3

2,6,18

5x-2 = 18

5x = 20

x = 4

Answer 3

For that to be a geometric sequence, the next term in the sequence must be a constant multiple

of the current term, so since a₁ = the first term in the sequence = 2 and a₂ = the second term in

the sequence = 6, then the multiple that we need is a₂ / a₁ = 6 / 2 = 3, so a₃ = the third term in

the sequence must be = 3 * a₂ = 3 * 6 = 18, but by what we are given a₃ also = 5x - 2,

so we have that 5x - 2 = 18, so 5x = 20, so x = 4.

Answer 4

5x - 2 = 6/2·6; 5x - 2 = 18; x = 4