Question

The 4th term of a gp series is 24 and 8th term is 384 find the series

Answer 1

Let the series be a,ar,ar^2,ar^3.....

We know that nth term of the GP is ar^(n - 1).

= > 4th term of GP = ar^(4 - 1) = ar^3.

ar^3 = 24 ----- (1)

= > 8th term of GP = ar^(8 - 1) = ar^7.

ar^7 = 384 ----- (2)

Now,

= > ar^7/ar^3 = 384/24

= > r^4 = 16

= > r^4 = (2)^4

= > r = 2.

Substitute r = 4 in (1), we get

= > ar^3 = 24

= > a(2)^3 = 24

= > a * 8 = 24

= > a = 24/8

= > a = 3.

Therefore, the series = 3, 6, 12,24,....

Hope this helps!

Answer 2

$\Large{\textbf{\underline{\underline{According\:to\:the\:Question}}}}$

Assumption

$\textbf{\underline{Geometric\;Progression}}$

= a , ar , ar² .....

Where,

nth term will be :-

${\boxed{\sf\:{T_{n}=ar^{n-1}}}}$

Hence,

ar³ = 24 ......(1)

${\boxed{\sf\:{ar_{7}=384....(2)}}}$

Dividing (2) by (1) we get :-

${\boxed{\sf\:{\dfrac{ar_{7}}{ar^3}=\dfrac{384}{24}}}}$

r⁴ = 16

We may also write it as :-

r⁴ = 2⁴

Power will be cancelled :-

r = 2

Hence,

Common ratio = 2

Substitute value of in (1)

ar³ = 24

a × 2³ = 24

8a = 24

${\boxed{\sf\:{a=\dfrac{24}{8}}}}$

a = 3

Therefore,

First term = 3

Series are follows :-

3

3 × 2

(3 × 2²)

(3 × 2³)

(3 × 2⁴)

= 3, 6, 12 , 24 , 48