Answer:
- ratio: 1/2
- first terms: 4, 2, 1, ...
Explanation:
You want the common ratio and first 3 terms of a decreasing geometric progression with the sum of the first three terms being 7, and their product being 8.
Setup
Let the first term be represented by x, and let r represent the common ratio. Then the first three terms are ...
x, xr, xr²
Their sum is ...
7 = x +xr +xr²
Their product is ...
8 = (x)(xr)(xr²) = (xr)³
Solution
Taking the cube root of the product equation, we have ...
2 = xr
Substituting this into the first equation, we have ...
7 = x +2 + 2r
5 = x +2r ⇒ x = 5 -2r
And substituting back into the above, we get ...
2 = (5 -2r)(r)
2r² -5r +2 = 0
(2r -1)(r -2) = 0
r = 2 or 1/2
We want r < 1, so r = 1/2.
x = 5 -2(1/2) = 4
Progression
For x = 4, r = 1/2, the first three terms are ...
x, xr, xr² = 4, 2, 1
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Additional comment
The equations are nicely solved by a graphing calculator. In the attached, we used y instead of r. We want the solution with y<1.
The two solutions give rise to terms 4, 2, 1 (decreasing) or 1, 2, 4 (increasing).