Illustrate a chalkboard with several mathematical equations written on it. Include a geometric sequence with its second term labeled as 6, and show the sum of the sequence as 27. Also, depict a written formula 'u1=6/r' and the formula for the sum of an infinite geometric series 'sum infinity = u1/1-r, [r]<1'. Exclude any text explanation or question, focusing solely on the mathematical demonstration.

"The sum of an infinite geometric sequence is 27. The second term of the sequence is 6. Find the possible values of r."

I made a formula 'u1=6/r' using the info. given above, and I was going to insert it into the formula for the sum of an infinite geometric series 'sum infinity = u1/r-1, [r]<1'. But how do you know if r is less than 1 or greater than 1? The markscheme used u1/1-r, how did they know?

Thanks in advance.

The sequence converges, so you know that |r|<1

But that's not really an issue here.
a/(1-r) = 27
(6/r)/(1-r) = 27
6 = 27r(1-r)
27r^2 - 27r + 6 = 0
3(3r-1)(3r-2) = 0
r = 1/3 or 2/3

oobleck, when does a sequence converge or diverge?

think about it. The sequence is

a, ar, ar^2, ...
if r>1 the terms keep getting bigger and bigger.
If r=1, the sequence is constant, so the sum will diverge (unless a=0)
So for a GP to converge, |r| < 1

That makes sense. But which part in the question gives away that it is a convergent geometric sequence because they haven't mentioned r values? Sorry to keep asking.

The sum of an infinite geometric sequence is 27.

The fact that the sum of an infinite number of terms is a finite value is the definition of convergent. Maybe you should google the topic and do some further reading. You will find much more information than I can provide here.

Well, I must say that identifying if r is less than 1 or greater than 1 can be quite a tricky business, kind of like trying to catch a clown riding a unicycle while juggling bowling pins. But fear not, my friend, for I have a humorous way to help you figure it out.

You see, when it comes to geometric sequences, the common ratio r has a big impact on whether the sequence converges or diverges. If the absolute value of r is less than 1, then the sequence converges to a finite sum. If the absolute value of r is greater than 1, then the sequence diverges to infinity.

So here's the joke-ematic rule: if the sequence is like a well-behaved child who knows how to converge and behave itself, then r must be less than 1. But if the sequence is more like a wild and crazy party animal, dancing into infinity, then r must be greater than 1.

In your case, since the sum of the geometric sequence is 27, it means the sequence converges to a finite sum, and therefore, r must be less than 1. That's why the markscheme used the formula u1/1-r instead of u1/r-1.

So remember, when it comes to geometric sequences, r must learn to behave and stay less than 1 if it wants to join the convergence party!

To find the possible values of r in an infinite geometric sequence, we can use the formula for the sum of an infinite geometric series which is:

sum infinity = u1 / (1 - r)

Where:
- sum infinity represents the sum of the infinite geometric series
- u1 represents the first term of the series
- r represents the common ratio of the series

Given that the sum of the infinite geometric series is 27, and the second term (u2) is 6, we can write the relationship between these values as:

27 = u1 / (1 - r) ...(1)
6 = u1 * r ...(2)

To solve for the possible values of r, we can substitute equation (2) into equation (1), since both equations involve u1:

27 = (6 / r) / (1 - r)

Simplifying the right side:

27 = 6 / (r - r^2)

Multiply both sides by (r - r^2) to eliminate the denominator:

27(r - r^2) = 6

Expanding the equation:

27r - 27r^2 = 6

Rearranging the terms:

27r^2 - 27r + 6 = 0

Now we have a quadratic equation. We can solve for r by factoring or using the quadratic formula. After solving, we find two possible values for r:

r = 1/3 or r = 2

Therefore, the possible values of r in this case are 1/3 and 2.

Regarding the markscheme using u1 / (1 - r) instead of u1 / (r - 1), it could be due to a difference in convention or notation. In most textbooks and resources, the formula for the sum of an infinite geometric series is given as u1 / (1 - r). However, some sources might use a different notation for the same formula, such as u1 / (r - 1). Both notations are correct as long as the relationship between the terms and the common ratio is correctly expressed in the equation.