Question

Given T5 = 96 and T8 = 768 of a geometric progression. Find the first term,a and the common ratio,r.

Answer 1

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:

$\displaystyle T_ n = ar^{n-1}$

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:

$\displaystyle T_5 = a r^{(5) - 1} \text{ and } T_8 = ar^{(8)-1}$

Substitute and simplify:

$\displaystyle 96 = ar^4 \text{ and } 768 = ar^7$

We can rewrite the second equation as:

$\displaystyle 768 = (ar^4) \cdot r^3$

Substitute:

$\displaystyle 768 = (96) r^3$

Hence:

$\displaystyle r = \sqrt[3]{\frac{768}{96}} = \sqrt[3]{8} = 2$

So, the common ratio r is two.

Using the first equation, we can solve for the initial term:

$\displaystyle \begin{aligned} 96 &= ar^4 \\ ar^4 &= 96 \\ a(2)^4 &= 96 \\ 16a &= 96 \\ a &= 6 \end{aligned}$

In conclusion, of the given geometric sequence, the first term a is 6 and its common ratio r is 2.