u = sum of an arithmetic series, 4 terms, common difference = 3
v = sum of a geometric series, 4 terms, multiplier = 3
Let it be known that u - v = 162
For your added pleasure, 1st terms are the same
Your mission: find the 1st term.
Hello, DenisB!
$\displaystyle u$ = sum of an arithmetic series, 4 terms, common difference = 3
$\displaystyle v$ = sum of a geometric series, 4 terms, common ratio = 3
Let it be known that: $\displaystyle u - v \:=\: 162$
For your added pleasure, 1st terms are the same
Your mission: find the 1st term.
Let $\displaystyle a$ = first term.
$\displaystyle u \;=\;\sum \begin{Bmatrix} a \\ a+3 \\ a+6 \\ a+9 \end{Bmatrix}\quad\Rightarrow\quad u \:=\:4a+18$
$\displaystyle v \;=\;\sum \begin{Bmatrix} a \\ 3a \\ 9a \\ 27a \end{Bmatrix} \quad\Rightarrow\quad v \:=\:40a$
$\displaystyle u-v \:=\:162 \quad\Rightarrow\quad (4a+18) - 40a \:=\:162$
. . . $\displaystyle -36a \;=\;144 \quad\Rightarrow\quad \boxed{a \;=\;-4}$