What is the sum of the geometric sequence 1, 3, 9, ... if there are 14 terms?
What is the sum of the geometric sequence 1, 3, 9, ... if there are 14 terms?
$$$$
$$\small{\text{
geometric sequence: $a_1 = 1 \quad r = 3$
}}\\
\small{\text{
$
\begin{array}{lcll}
\hline
\\
s_{14} &=& \textcolor[rgb]{150,0,0}{1} + & 3^1 + 3^2 + 3^3
+ 3^4 + 3^5 + 3^6
+ 3^7 + 3^8 + 3^9
+ 3^{10} + 3^{11} + 3^{12} + 3^{13} \\
3\cdot s_{14} &=& & 3^1 + 3^2 + 3^3
+ 3^4 + 3^5 + 3^6
+ 3^7 + 3^8 + 3^9
+ 3^{10} + 3^{11} + 3^{12} + 3^{13} + \textcolor[rgb]{150,0,0}{3^{14}}\\
\hline
\\
s_{14}-3\cdot s_{14} &=& \textcolor[rgb]{150,0,0}{1-}&\textcolor[rgb]{150,0,0}{3^{14}}\\
s_{14}\cdot(1-3) &=& 1- &3^{14}\\
-2\cdot s_{14} &=& 1- &3^{14}\\
\end{array}
$}}$$
$$\small{\text{
$
\begin{array}{rcl}
s_{14} & = & \dfrac { 1- 3^{14} } {-2}\\\\
s_{14} & = & \dfrac { 3^{14}-1 } {2} = 2\,391\,484 \\
\end{array}
$
}}$$
What is the sum of the geometric sequence 1, 3, 9, ... if there are 14 terms?
$$$$
$$\small{\text{
geometric sequence: $a_1 = 1 \quad r = 3$
}}\\
\small{\text{
$
\begin{array}{lcll}
\hline
\\
s_{14} &=& \textcolor[rgb]{150,0,0}{1} + & 3^1 + 3^2 + 3^3
+ 3^4 + 3^5 + 3^6
+ 3^7 + 3^8 + 3^9
+ 3^{10} + 3^{11} + 3^{12} + 3^{13} \\
3\cdot s_{14} &=& & 3^1 + 3^2 + 3^3
+ 3^4 + 3^5 + 3^6
+ 3^7 + 3^8 + 3^9
+ 3^{10} + 3^{11} + 3^{12} + 3^{13} + \textcolor[rgb]{150,0,0}{3^{14}}\\
\hline
\\
s_{14}-3\cdot s_{14} &=& \textcolor[rgb]{150,0,0}{1-}&\textcolor[rgb]{150,0,0}{3^{14}}\\
s_{14}\cdot(1-3) &=& 1- &3^{14}\\
-2\cdot s_{14} &=& 1- &3^{14}\\
\end{array}
$}}$$
$$\small{\text{
$
\begin{array}{rcl}
s_{14} & = & \dfrac { 1- 3^{14} } {-2}\\\\
s_{14} & = & \dfrac { 3^{14}-1 } {2} = 2\,391\,484 \\
\end{array}
$
}}$$