Geomtrice - Arithmetic Sequence Problem

Three terms of a sequence form a geometric sequence and add to 39. If the middle term is increased by a factor of  
\[\frac{5}{3}\]
  these three terms form successive terms in an arithmetic sequence. What are the possible values of the first term?
Let the three terms be  
\[a, \: ar, \: ar^2\]
,
The arithmetic sequence is  
\[a, \: \frac{5}{3}ar, \: ar^2\]
.
Since these terms are successive term in an arithmetic sequence,  
\[ar^2- \frac{5}{3}=\frac{5}{3}-a \rightarrow 3r^2-10r+3=0 \rightarrow (r-3)(3r-1) \rightarrow r=3, \: \frac{1}{3}\]
.
If  
\[r=3\]
,  
\[a+ar+ar^2=39 \rightarrow 13a=39 \rightarrow a=3\]
.
If  
\[r=\frac{1}{3}\]
,  
\[a+ar+ar^2=39 \rightarrow \frac{13}{9}a=39 \rightarrow a=27\]
.

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