SOLUTION: can you solve this without log? if yes, how do u solve this? a sequence t1, t2, t3, ... is defined by t1= 1000 and tn= 1.06tn-1, where n is a positive integer and n>_ 2. find th

Algebra ->  Test -> SOLUTION: can you solve this without log? if yes, how do u solve this? a sequence t1, t2, t3, ... is defined by t1= 1000 and tn= 1.06tn-1, where n is a positive integer and n>_ 2. find th      Log On


   

Question 373821: can you solve this without log? if yes, how do u solve this?
a sequence t1, t2, t3, ... is defined by t1= 1000 and tn= 1.06tn-1, where n is a positive integer and n>_ 2. find the smallest value of n such that tn>3000
Answer by robertb(5830) About Me  (Show Source):
You can put this solution on YOUR website!
Consider
t(1) = 1,000 and t(n) = 1.06t(n-1). The 2nd equation gives t%28n%29%2Ft%28n-1%29+=+1.06. Then %28t%28n%29%2Ft%28n-1%29%29%2A%28t%28n-1%29%2Ft%28n-2%29%29*...*. Hence
t%28n%29%2F1000+=+1.06%5E%28n-1%29, or t%28n%29+=+1000%2A1.06%5E%28n-1%29.
We want 1000%2A%281.06%29%5E%28n-1%29%3E3000, or %281.06%29%5E%28n-1%29%3E3.
Take ln of both sides: %28n-1%29ln1.06%3Eln3, or n-1+%3E+ln3%2Fln1.06+=+18.8542. Thus n > 19.8542, so that n%3E=20.