The first, second and third terms of a geometric progression are 2k+3, k+6 and k, respectively. Given that all the terms of geometric progression are positive, calculate

(a) the value of the constant k
(b) the sum to infinity of the progression.

since the ratio between terms is constant,

(k+6)/(2k+3) = k/(k+6)
(k+6)^2 = k(2k+3)
k^2+12k+36 = 2k^2+3k
k^2 - 9k - 36 = 0
(k-12)(k+3) = 0
k = 12

and the sequence is
27,18,12,...

so the ratio is 2/3 and S = 27/(1/3) = 81

IF THE 6TH TERM OF A SERIES IS 24 AND THE COMMON DIFFERENCE IS 3 FIND THE SERIES

(a) Well, let's start by looking at the first and second terms. In a geometric progression, if you divide any term by the previous term, you should get a constant ratio. So, let's find the ratio between the second and first terms:

Ratio = (k + 6) / (2k + 3)

Now, let's find the ratio between the third and second terms:

Ratio = k / (k + 6)

Since all the terms are positive, we can set the two ratios equal to each other and solve for k:

(k + 6) / (2k + 3) = k / (k + 6)

Cross-multiplying, we get:

k^2 + 6k = (2k + 3)(k)

Expanding the right side, we get:

k^2 + 6k = 2k^2 + 3k

Rearranging and simplifying, we have:

k^2 - 3k = 0

Factoring out k, we get:

k(k - 3) = 0

So, either k = 0 or k - 3 = 0. However, since we're looking for positive terms, k can only be 3. Therefore, k = 3.

(b) Now, to find the sum to infinity of the geometric progression, we use the formula:

Sum = a / (1 - r)

where 'a' is the first term and 'r' is the common ratio.

In this case, a = 2k + 3 and r = (k + 6) / (2k + 3).

So, the sum to infinity becomes:

Sum = (2k + 3) / (1 - (k + 6) / (2k + 3))

Substituting k = 3, we get:

Sum = (2(3) + 3) / (1 - (3 + 6) / (2(3) + 3))

Sum = (6 + 3) / (1 - 9 / 9)

Sum = 9 / (1 - 1)

Sum = 9 / 0

And there you have it, the sum to infinity of this geometric progression is undefined! It's like trying to divide a pie by zero - it just doesn't work. Sorry to break the mathematical rules, but infinity has got a sneaky sense of humor sometimes. Keep smiling! 🤡

To find the value of the constant k and the sum to infinity of the geometric progression, we can use the following steps:

Step 1: Determine the common ratio (r):
The common ratio (r) of a geometric progression is found by dividing any term by its preceding term. So, in this case, we have:
r = (k + 6) / (2k + 3)

Step 2: Set up an equation using the given terms:
Using the given terms, we can set up the equation as follows:
(2k + 3) / (k + 6) = (k + 6) / (k)

Step 3: Solve the equation:
To solve this equation, we can cross-multiply and simplify:
(k + 6)(k) = (k + 6)(2k + 3)
k^2 = (2k + 3)(k + 6)
k^2 = 2k^2 + 15k + 18
Rearranging and simplifying:
k^2 - 15k - 18 = 0

Step 4: Solve for k:
We can now solve this quadratic equation to find the value of k. Factoring or using the quadratic formula, we get:
(k - 18)(k + 1) = 0
Solving for k, we have two possible solutions:
k - 18 = 0 or k + 1 = 0
k = 18 or k = -1

However, since all the terms of the geometric progression need to be positive, we discard the solution k = -1.

(a) The value of the constant k is 18.

Step 5: Find the sum to infinity:
The sum to infinity of a geometric progression can be calculated using the formula S = a / (1 - r), where 'a' is the first term and 'r' is the common ratio.

In this case, the first term, a, is 2k + 3 and the common ratio, r, is (k + 6) / (2k + 3). So, to find the sum to infinity, we substitute these values into the formula:
S = (2k + 3) / (1 - (k + 6) / (2k + 3))

(b) Calculate the sum to infinity using the found value of k.

Substituting k = 18 into the formula, we get:
S = (2 * 18 + 3) / (1 - (18 + 6) / (2 * 18 + 3))
S = 39 / (1 - 24 / 39)
S = 39 / (15 / 39)
S = (39 * 39) / 15
S = 101.4

(b) The sum to infinity of the geometric progression is 101.4.