Answer:
Please check the explanation.
Step-by-step explanation:
Given the geometric sequence
a – 2, a, a + 4
- We know that the common ratio 'r' of a geometric sequence can be obtained by dividing the successor term by the previous term.
- Also, we know that the common ratio 'r' of a geometric sequence is the same i.e. remain constant.
so the expression to find the common ratio 'r' of the geometric sequence will be:
[tex]\:r[/tex] ⇒ [tex]\frac{a}{a-2}=\frac{a+4}{a}[/tex]
[tex]\frac{a}{a-2}=\frac{a+4}{a}[/tex]
[tex]\mathrm{Apply\:fraction\:cross\:multiply:\:if\:}\frac{a}{b}=\frac{c}{d}\mathrm{\:then\:}a\cdot \:d=b\cdot \:c[/tex]
[tex]aa=\left(a-2\right)\left(a+4\right)[/tex]
[tex]a^2=\left(a-2\right)\left(a+4\right)[/tex]
[tex]a^2=a^2+2a-8[/tex]
[tex]\mathrm{Subtract\:}a^2+2a\mathrm{\:from\:both\:sides}[/tex]
[tex]a^2-\left(a^2+2a\right)=a^2+2a-8-\left(a^2+2a\right)[/tex]
[tex]-2a=-8[/tex]
[tex]\mathrm{Divide\:both\:sides\:by\:}-2[/tex]
[tex]\frac{-2a}{-2}=\frac{-8}{-2}[/tex]
[tex]a=4[/tex]
so the ratio becomes
[tex]r=\:\frac{4}{4-2}=\frac{4}{2} =2[/tex]
Hence, the common ratio 'r' will be:
[tex]r=2[/tex]
VERIFICATION
so the sequence becomes
a – 2, a, a + 4
(4) - 2, 4, (4)+4 ∵ a=4
2, 4, 8
From the sequence it is clear that
4/2=8/4 ⇒ r
2=2 ⇒ r
Hence, the common ration 'r=2' is the same.
Therefore, the common ratio 'r' will be:
[tex]r=2[/tex]
