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

3rd term = a r² = 12

6th term = a r⁵ = 96

(a r⁵) / (a r²) = r³ = 96/ 12 = 8 => r = 2

Since a r² = 12 => a = 12 / r² = 3

So the series is : 3, 6, 12, 24, 48, 96, 192, 384, 768, 1536. (we can simply count the number of terms here.). Or, we can find it mathematically as below.

nth term = a

Let us find the highest n such that nth term is less than 2000.

. 3 *

< 2000

=>

< 2000/3 = 666.66..

=>

< 666.66..

We have

= 512 and

= 1024.

So the highest term less than 2000 is for n = 9.

Hence,

*there are n+1 terms* less than 2000

*: so *__10 terms.__=================================

We can also find it by :

Log

< log (2000/3)

n log 2 < log (2000/3)

n < [ log (2000/3) ] / Log 2

n < 9.28

So n = 9.

Hence there are

** 10 terms i**n the GP that are less than 2000.