Question

if the third and sixth terms of gp are 12 and 96 .find the number of terms less than 2000

Answer 1

Heya....

Here's your answer....

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 r^n

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

 .     3 * 2^n < 2000

 =>     2^n < 2000/3 = 666.66..

=>      2^n < 666.66..

 We have 2^9 = 512    and    2^{10}  = 1024.

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

Hence, there are n+1 terms less than 2000:  so 10 terms.

Thanks...!!!

XD

Sorry baby 'wink'