Answer:
[tex]r=\frac{1}{4}\\\\r=-\frac{5}{4}[/tex]
Step-by-step explanation:
Geometric Sequences
The geometric sequence is given as:
[tex]a_1, a_1r, a_1r^2, a_1r^3,..., a_1r^{n-1}[/tex]
Where n is the number of the term, n≥1, and r is the common ratio.
The sum of n terms of the geometric sequence is given by:
[tex]\displaystyle S_n=a_1\frac{r^n-1}{r-1}[/tex]
We are given: S3=252, a1=192, thus substuting:
[tex]\displaystyle 252=192\frac{r^3-1}{r-1}[/tex]
Dividing by 12:
[tex]\displaystyle 21=16\frac{r^3-1}{r-1}[/tex]
Recall that:
[tex]r^3-1=(r-1)(r^2+r+1)[/tex]
Substituting:
[tex]\displaystyle 21=16\frac{(r-1)(r^2+r+1)}{r-1}[/tex]
Simplifying:
[tex]21=16(r^2+r+1)=16r^2+16r+16[/tex]
Rearranging:
[tex]16r^2+16r+16-21=0[/tex]
16r^2+16r-5=0
Rewriting:
[tex]16r^2-4r+20r-5=0[/tex]
Factoring:
[tex]4r(4r-1)+5(4r-1)=0[/tex]
[tex](4r-1)(4r+5)=0[/tex]
Solving:
[tex]r=\frac{1}{4}\\\\r=-\frac{5}{4}[/tex]
Both solutions are valid
