Question

the 4th and 8th terms of a G.P. are 24 and 384 respectively. find the first term and the common ratio

Answer 1

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Answer 2

The first term(a) = 3 and the common ratio(r) = 2

Step-by-step explanation:

Given,

$a_{4} =24$ and $a_{8} =384$

To find, the first term and the common ratio = ?

Let the first term = a and the common ratio = r

We know that,

nth terms of a G.P.

$a_{n} =ar^{n-1}$

∴ $a_{4} =ar^{4-1}=ar^{3}$

⇒ $ar^{3}=24$    ....(1)

Also,

$a_{8} =ar^{8-1}=ar^{7}$

⇒ $ar^{7}=384$  

⇒$(ar^{3})r^{4} =384$  

Using (1), we get

$24\times r^{4} =384$

⇒ $r^{4} =\dfrac{384}{24} =16=2^{4}$

∴ r = 2

Putting r = 2 in (1), we get

$a2^{3}=24$

⇒ a = $\dfrac{24}{8} =3$

Hence, the first term(a) = 3 and the common ratio(r) = 2