Forum
Forums
Questions and Answe...
Algebra
Formulas
Sequences
 
Share:
Notifications
Clear all

[Solved] Sequences

 
Formulas
Last Post by Staff 4 years ago
1 Posts
1 Users
0 Reactions
288 Views
RSS
0
June 15, 2021 10:48 am
Topic starter
Staff 
(@armor)
Admin
  Honorable Member
317 Posts
156 153 0

The sequence  \left \{a _{n} \right \} is a geometric sequence, a_{1}=8,    b_{n}=log_{2}a_{n}.

If the first 7th partial sum  S_{7}  of  \left \{ b_{n} \right \}  is the maximum value, and S_{7}\neq S_{8}. 

Find the range of the common ratio q of the sequence \left \{ a_{n} \right \}.

Add a comment
1 Answer
0
June 15, 2021 10:48 am
Topic starter
Staff 
(@armor)
Admin
  Honorable Member
317 Posts
156 153 0

\small b_{n+1}-b_{n}=log_{2} a_{n+1}-log_{2}a_{n}=log_{2}\frac{a_{n+1}}{a_{n}}=log_{2}q

Then \small \left \{ b_{n} \right \} is an arithmetic sequence, and its first term is \small b_{1}=log_{2}a_{1}=3,   

its common difference is \small log_{2}q.  

From the given condition, we have \small b_{7}\geq 0, b_{8}< 0.

Then 

\small 3+6log_{2}q\geq 0

\small 3+7log_{2}q<0

Thus \small -\frac{1}{2}\leq log_{2}q<-\frac{3}{7}.

Hence \small q\in [\frac{\sqrt{2}}{2},2^{-\frac{3}{7}}).

Add a comment
Forum Jump:
  Previous Topic
Next Topic  
Share: