Answer:
a = 24 and r = 2
Step-by-step explanation:
The n th term of a geometric progression is
[tex]a_{n}[/tex] = a[tex](r)^{n-1}[/tex], thus
[tex]a_{6}[/tex] = a[tex]r^{5}[/tex] = 768 → (1) and
a₃ = a[tex](6r)^{2}[/tex]
Simplifying gives
a[tex]r^{5}[/tex] = 768 → (3)
36ar² = 3456 → (4)
Divide (3) by (4)
[tex]\frac{r^3}{36}[/tex] = [tex]\frac{768}{3456}[/tex] = [tex]\frac{2}{9}[/tex]
Cross- multiply
9r³ = 72 ( divide both sides by 9 )
r³ = 8 ( take the cube root of both sides )
r = [tex]\sqrt[3]{8}[/tex] = 2
Substitute r = 2 into (3)
a[tex]2^{5}[/tex] = 768, that is
32a = 768 ( divide both sides by 32 )
a = 24
