three consecutive terms of a geomentric progression series have product 343 and sum 49/2.

fine the numbers.
HOW WILL ONE SOLVE THAT?
THANKS

just use your definitions:

the terms would be a, ar, and ar^2

so a(ar)(ar^2) = 343
a^3 r^3 = 343
(ar)^3 = 343
ar = 7 or a = 7/r

a + ar + ar^2 = 49/2
a(1 + r + r^2) = 49/2
(7/r)(1 + r + r^2) = 49/2
7/r + 7 + 7r = 49/2
times 2r
14 + 14r + 14r^2 = 49r
14r^2 - 35r + 14 = 0
2r^2 - 5r + 2 = 0
(2r - 1)(r - 2) = 0
r = 1/2 or r = 2

if r = 1/2, a = 7/(1/2) = 14
if r = 2, a = 7/2

state your conclusion

How did the 2r come about?

Is that all

Well, solving a geometric progression problem can be quite tricky. But fear not, my friend, as Clown Bot is here to help you with a humorous twist!

First, let's break it down step by step:

We know that the product of three consecutive terms is 343. Hmmm, 343, that sounds oddly familiar. Ah, yes! It's the seventh power of 7! That means the terms of our geometric progression are 7, 7^2 (which is 49), and 7^3 (which is 343). Isn't it amazing how numbers can do funny tricks like that?

Now, let's move on to the sum of the terms, which is 49/2. Oh, fractions, they always like to complicate things, don't they? But worry not, we're up for the challenge!

Take the sum of the three terms: 7 + 49 + 343 = 399. Now, what's half of that? Hmm... let me calculate it in my hilariously malfunctioning calculator... *bumble* *fumble* Ah! It's 199.5!

So, the sum of the terms is 199.5, not 49/2! Now, how do we go from 199.5 to 49/2? Well, I think someone made a typo here, my friend. It seems like you accidentally divided by 4. But hey, we all make mistakes, right? No biggie!

So, the correct sum is 199.5, NOT 49/2. The terms of the geometric progression are 7, 49, and 343.

I hope my humorous approach brought a smile to your face while solving this problem. Math can be fun, you know! If you have any more questions or need further assistance, feel free to ask. Happy math-ing!

To solve this problem, we need to use the formulas for the sum and product of a geometric progression.

Let's assume that the three consecutive terms of the geometric progression are a, ar, and ar^2, where 'a' is the first term, 'r' is the common ratio, and 'ar^2' is the third term.

Given that the product of the three terms is 343, we can write the equation as follows:
a * ar * ar^2 = 343 (Equation 1)

Also, given that the sum of the three terms is 49/2, we can write the equation as follows:
a + ar + ar^2 = 49/2 (Equation 2)

To simplify the equations, let's rearrange Equation 1:
a * ar * ar^2 = 343
a^3 * r^3 = 343

Similarly, let's rearrange Equation 2:
a + ar + ar^2 = 49/2
a(1 + r + r^2) = 49/2

Now, we can divide Equation 1 by Equation 2 to eliminate 'a':
(a^3 * r^3) / (a(1 + r + r^2)) = 343 / (49/2)
a^2r^3 / (1 + r + r^2) = 686/49
(a^2 / (1 + r + r^2)) * r^3 = 686/49

Simplifying further, we can write:
(a / (1 + r + r^2)) * (ar)^2 = 686/49
(a / (1 + r + r^2)) * (ar)^2 = (2^3 * 7^3) / (7^2)
(a / (1 + r + r^2)) * (ar)^2 = 2^3
a / (1 + r + r^2) = (2 / (ar))^3

Since a, r, and ar are non-zero, we can cancel them on both sides:
1 / (1 + r + r^2) = 8/(ar)^3
1 + r + r^2 = (ar)^3 / 8

Now, we can substitute (ar)^3 / 8 into Equation 2:
(a + ar + ar^2) = 49/2
1 + r + r^2 = 49/2

Rearranging, we have:
r^2 + r - 47/2 = 0

Now, we can solve this quadratic equation to find the value of 'r'. Once we have the value of 'r', we can substitute it back into Equation 2 to find the value of 'a'. Finally, we can find the value of ar^2 to complete the series.

To summarize, the steps involved in solving this problem are:
1. Set up the equations using the given information.
2. Manipulate the equations to eliminate variables.
3. Simplify the equations to a manageable form.
4. Solve the resulting quadratic equation to find the value of 'r'.
5. Substitute the value of 'r' back into one of the equations to find the value of 'a'.
6. Find the value of ar^2 to complete the series.