SOLUTION: The sum of the first three terms in a GP is 38. Their product is 1728. Find the values of the three terms. My answer were 8 and 18. I want to confirm if it's correct and how w

Question 1157937: The sum of the first three terms in a GP is 38. Their product is 1728. Find the values of the three terms.

My answer were 8 and 18. I want to confirm if it's correct and how will I find the values of the three terms with two solution of a and r
Found 2 solutions by ikleyn, MathTherapy:
Answer by ikleyn(52787) (Show Source): You can put this solution on YOUR website!
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Let "a" be the first term and "r" be the common ratio. Then from the condition, we have these two equations a + ar + ar^2 = 38, (1) a*(ar*)*(ar^2) = 1728. (2) From equation (2), a^3*r^3 = 1728, or (ar)^3 = 1728, which implies ar = = 12; (3) hence, r = . (4) Now, in equation (1) replace the term ar by 12, based on (3). You will get a + 12 + ar^2 = 38, which implies a + ar^2 = 26. (5) Next, substitute r = into equation (5), replacing "r" there. You will get + = 26, or + = 26. Multiply by "a" both sides and simplify a^2 - 26a + 144 = 0, - 169 + 144 = 0 = 25 a - 13 = +/- = +/- 5. Thus two solutions for "a" are a = 13 + 5 = 18 or a = 13 - 5 = 8. If a = 8, then from (4) r = = . If a = 18, then from (4) r = = . In the first case, if a = 8, then the three terms are 8, = 12 and = 18. In this case, the sum of terms is 8 + 12 + 18 = 38, so this solution does work. In the second case, if a = 18, then the three terms are 18, = 12 and = 8. In this case, the sum of terms is 18 + 12 + 8 = 38, so this solution does work, too. ANSWER. The problem has two solution: a) first term is 18; the common difference is and the progression is 18, 12, 8. b) first term is 8; the common difference is and the progression is 8, 12, 18.

Solved.

Answer by MathTherapy(10552) (Show Source): You can put this solution on YOUR website!

The sum of the first three terms in a GP is 38. Their product is 1728. Find the values of the three terms.
My answer were 8 and 18. I want to confirm if it's correct and how will I find the values of the three terms with two solution of a and r

SUM of the 3 terms:
------- eq (i)
Product of the 3 terms: , and so: ======>
--------- eq (ii)
We then get:
------ Cross-multiplying
------- Factoring out GCF, 2

3r(2r - 3) - 2(2r - 3) = 0
(3r - 2)(2r - 3) = 0
3r - 2 = 0 OR 2r - 3 = 0
3r = 2 OR 2r = 3

Therefore, if r, or common ratio = , then or 1st term = 18
In this case, the 3 terms are:

Therefore, if r, or common ratio = , then or 1st term = 8
In this case, the 3 terms are:

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