Answer:
Of the given geometric sequence, the first term a is 6 and its common ratio r is 2.
Step-by-step explanation:
Recall that the direct formula of a geometric sequence is given by:
[tex]\displaystyle T_ n = ar^{n-1}[/tex]
Where Tₙ is the nth term, a is the initial term, and r is the common ratio.
We are given that the fifth term T₅ = 96 and the eighth term T₈ = 768. In other words:
[tex]\displaystyle T_5 = a r^{(5) - 1} \text{ and } T_8 = ar^{(8)-1}[/tex]
Substitute and simplify:
[tex]\displaystyle 96 = ar^4 \text{ and } 768 = ar^7[/tex]
We can rewrite the second equation as:
[tex]\displaystyle 768 = (ar^4) \cdot r^3[/tex]
Substitute:
[tex]\displaystyle 768 = (96) r^3[/tex]
Hence:
[tex]\displaystyle r = \sqrt[3]{\frac{768}{96}} = \sqrt[3]{8} = 2[/tex]
So, the common ratio r is two.
Using the first equation, we can solve for the initial term:
[tex]\displaystyle \begin{aligned} 96 &= ar^4 \\ ar^4 &= 96 \\ a(2)^4 &= 96 \\ 16a &= 96 \\ a &= 6 \end{aligned}[/tex]
In conclusion, of the given geometric sequence, the first term a is 6 and its common ratio r is 2.
