## 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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