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

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 T 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.