What number must be added to each of the numbers 1, 3, and 6 in order for the resulting numbers to form a geometric sequence?
let the number to be added be x
then :
(3+x)/(1+x) = (6+x)/(3+x)
(1+x)(6+x) = (3+x)(3+x)
6 + 7x + x^2 = 9 + 6x + x^2
x = 3
check:
the sequence would then be 4,6, and 9
which is a GP with r = 3/2
How we get 7x and 6x??
Well, if you want to turn the numbers 1, 3, and 6 into a geometric sequence, you'll need a magical number that can transform them. Let's call this number the "Geometric Fairy Dust."
Now, before we sprinkle our Geometric Fairy Dust, let's see the pattern in these numbers. We notice that 3 divided by 1 is equal to 6 divided by 3. This means that the common ratio between the numbers is 2.
To maintain the geometric sequence, we need to add the same value to each number. So, if we sprinkle our Geometric Fairy Dust, it should cause an addition of the same number to each one.
Since the common ratio is 2, we can figure out the missing number by subtracting 1 from 3 (3 - 1 = 2) and subtracting 3 from 6 (6 - 3 = 3). Therefore, if we sprinkle our Geometric Fairy Dust, it should be the number 2 or 3, depending on how magical our dust is feeling.
Remember, though, that this is just a whimsical way to explain it. In reality, you would use a more mathematical approach to solve this problem.
To determine the number that needs to be added to each of the numbers 1, 3, and 6 in order for the resulting numbers to form a geometric sequence, we need to first understand the concept of a geometric sequence.
A geometric sequence is a sequence in which each term is found by multiplying the previous term by a fixed, non-zero number called the common ratio (r).
Let's denote the number we need to add to each of the given numbers as "x". So, our sequence becomes: 1 + x, 3 + x, and 6 + x.
For these numbers to form a geometric sequence, we need to have the ratio between any two consecutive terms (common ratio) be the same.
The common ratio (r) can be calculated by dividing any term by its previous term. So, let's set up the ratios:
(3 + x) / (1 + x) = (6 + x) / (3 + x)
Cross multiplying:
(3 + x) * (3 + x) = (6 + x) * (1 + x)
Expanding both sides:
9 + 3x + 3x + x^2 = 6 + 6x + x + x^2
Now, let's simplify the equation:
9 + 6x + x^2 = 6 + 6x + x + x^2
Simplifying further, we find that all the terms with "x" cancel out:
9 = 6
This equation is not true, which means that there is no value for "x" that will make the numbers 1 + x, 3 + x, and 6 + x form a geometric sequence.
Therefore, there is no specific number that needs to be added to each of the given numbers to form a geometric sequence.