Question

the first term of a geometric sequence is 8000 and the fifth term is 500. Determine the common ratio and the sum of the first nine terms​

Answer 1

Answer:

The common ratio  = 1/2

The sum of 9 terms of GP is 15,968.75

Step-by-step explanation:

Here, in the given GP:

Firs Term a = 8000, Fifth term a(5)  =   500

Let Common Ratio = r

Now, by the general term of GP:

$a_n = a \times (r)^{n-1}$

For, n = 5  $a_5 = a \times (r)^{5-1}$

or, $500 = 8000 \times (r)^{4}\\\implies (r)^{4} = \frac{500}{8000} = \frac{1}{16} = \frac{1}{(2)^4} \\\implies r= \frac{1}{2}$

Hence in the given GP, a   = 8000 and r = 1/2

Now, in a  GP sum of n terms is $s_n = \frac{a(1-r^n)}{1-r}$

So, for n = 9, $s_9 = \frac{8000(1- (\frac{1}{2}) ^9)}{1-\frac{1}{2} } = \frac{8000(1- 0.001953)}{0.5 }\\= \frac{8000 \times (0.9980)}{0.5} = 15,968.75$

So,the sum of 9 terms of GP is 15,968.75.