1. ## Series...

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

Your mission: find the 1st term.

2. ## Re: Series...

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: $\displaystyle u - v \:=\: 162$

Your mission: find the 1st term.

Let $\displaystyle a$ = first term.

$\displaystyle u \;=\;\sum \begin{Bmatrix} a \\ a+3 \\ a+6 \\ a+9 \end{Bmatrix}\quad\Rightarrow\quad u \:=\:4a+18$

$\displaystyle v \;=\;\sum \begin{Bmatrix} a \\ 3a \\ 9a \\ 27a \end{Bmatrix} \quad\Rightarrow\quad v \:=\:40a$

$\displaystyle u-v \:=\:162 \quad\Rightarrow\quad (4a+18) - 40a \:=\:162$

. . . $\displaystyle -36a \;=\;144 \quad\Rightarrow\quad \boxed{a \;=\;-4}$

3. ## Re: Series...

Perfecto! Can't fool you, Soroban.

-4, -1, 2, 5 : 2
-4,-12,-36,-108 : -160