Answer:
The value of the three terms is 8 and 18
Step-by-step explanation:
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 = root%283%2C1728%29 = 12; (3)
hence,
r = 12%2Fa. (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 = 12%2Fa into equation (5), replacing "r" there. You will get
a + a%2A%28144%2Fa%5E2%29 = 26, or
a + 144%2Fa = 26.
Multiply by "a" both sides and simplify
a^2 - 26a + 144 = 0,
%28a-13%29%5E2 - 169 + 144 = 0
%28a-13%29%5E2 = 25
a - 13 = +/- sqrt%2825%29 = +/- 5.
Thus two solutions for "a" are a = 13 + 5 = 18 or a = 13 - 5 = 8.
If a = 8, then from (4) r = 12%2F8 = 3%2F2.
If a = 18, then from (4) r = 12%2F18 = 2%2F3.
In the first case, if a = 8, then the three terms are 8, 8%2A%283%2F2%29 = 12 and 8%2A%283%2F2%29%5E2 = 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, 18%2A%282%2F3%29 = 12 and 18%2A%282%2F3%29%5E2 = 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 2%2F3 and the progression is 18, 12, 8.
b) first term is 8; the common difference is 3%2F2 and the progression is 8, 12, 18.
