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A geometric sequence has 400 terms. The first term is 1000 and the common ratio is -\frac{8}{9}$ How many terms of this sequence are greater than 1?

 Jun 16, 2024
 #1
avatar+129733 
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1000 (-8/9)^(n -1)  > 1

 

(-8/9)^(n -1)   is  only positive when  n  is odd

 

So we can  write

 

1000 (8/9)^(n -1)  >  1

 

(8/9)^(n - 1) > 1/1000     take the log of both sides  and by a log property we can write

 

(n -1) log (8/9) > log (1/1000)        

 

{ divide both sides by log (8/9)....this is negative so reverse the inequality sign }

 

n - 1  <   log (1/1000) / (log (8/9)

 

n <  1 + log (1/1000) /log (8/9)

 

n < 59.6

 

So 

 

n = 59

 

The number of terms > 1 =   [ 59 + 1 ] / 2  =  30

 

cool  cool  cool

 Jun 16, 2024

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