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

The common ratio = 1/2

The sum of 9 terms of GP is 15,968.75

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 * (r)^(n-1)`

For, n = 5
`a_5 = a * (r)^(5-1)`

or,
`500  = 8000 * (r)^(4)\\\implies  (r)^(4) = (500)/(8000)  = (1)/(16)   = (1)/((2)^4) \\\implies r= (1)/(2)`

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

Now, in a GP sum of n terms is
`s_n = (a(1-r^n))/(1-r)`

So, for n = 9,
`s_9 = (8000(1- ((1)/(2)) ^9))/(1-(1)/(2) ) = (8000(1- 0.001953))/(0.5 )\\= (8000 * (0.9980))/(0.5)  =  15,968.75`

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