\[\frac{5}{3}\]
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
\[r=\frac{1}{3}\]
, \[a+ar+ar^2=39 \rightarrow \frac{13}{9}a=39 \rightarrow a=27\]
.