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What is the sum of the geometric sequence 1, 3, 9, ... if there are 14 terms?

 Apr 29, 2015

Best Answer 

 #1
avatar+26400 
+5

What is the sum of the geometric sequence 1, 3, 9, ... if there are 14 terms?

$$$$

$$\small{\text{
geometric sequence: $a_1 = 1 \quad r = 3$
}}\\
\small{\text{
$
\begin{array}{lcll}
\hline
\\
s_{14} &=& \textcolor[rgb]{150,0,0}{1} + & 3^1 + 3^2 + 3^3
+ 3^4 + 3^5 + 3^6
+ 3^7 + 3^8 + 3^9
+ 3^{10} + 3^{11} + 3^{12} + 3^{13} \\
3\cdot s_{14} &=& & 3^1 + 3^2 + 3^3
+ 3^4 + 3^5 + 3^6
+ 3^7 + 3^8 + 3^9
+ 3^{10} + 3^{11} + 3^{12} + 3^{13} + \textcolor[rgb]{150,0,0}{3^{14}}\\
\hline
\\
s_{14}-3\cdot s_{14} &=& \textcolor[rgb]{150,0,0}{1-}&\textcolor[rgb]{150,0,0}{3^{14}}\\
s_{14}\cdot(1-3) &=& 1- &3^{14}\\
-2\cdot s_{14} &=& 1- &3^{14}\\
\end{array}
$}}$$

$$\small{\text{
$
\begin{array}{rcl}
s_{14} & = & \dfrac { 1- 3^{14} } {-2}\\\\
s_{14} & = & \dfrac { 3^{14}-1 } {2} = 2\,391\,484 \\
\end{array}
$
}}$$

 Apr 29, 2015
 #1
avatar+26400 
+5
Best Answer

What is the sum of the geometric sequence 1, 3, 9, ... if there are 14 terms?

$$$$

$$\small{\text{
geometric sequence: $a_1 = 1 \quad r = 3$
}}\\
\small{\text{
$
\begin{array}{lcll}
\hline
\\
s_{14} &=& \textcolor[rgb]{150,0,0}{1} + & 3^1 + 3^2 + 3^3
+ 3^4 + 3^5 + 3^6
+ 3^7 + 3^8 + 3^9
+ 3^{10} + 3^{11} + 3^{12} + 3^{13} \\
3\cdot s_{14} &=& & 3^1 + 3^2 + 3^3
+ 3^4 + 3^5 + 3^6
+ 3^7 + 3^8 + 3^9
+ 3^{10} + 3^{11} + 3^{12} + 3^{13} + \textcolor[rgb]{150,0,0}{3^{14}}\\
\hline
\\
s_{14}-3\cdot s_{14} &=& \textcolor[rgb]{150,0,0}{1-}&\textcolor[rgb]{150,0,0}{3^{14}}\\
s_{14}\cdot(1-3) &=& 1- &3^{14}\\
-2\cdot s_{14} &=& 1- &3^{14}\\
\end{array}
$}}$$

$$\small{\text{
$
\begin{array}{rcl}
s_{14} & = & \dfrac { 1- 3^{14} } {-2}\\\\
s_{14} & = & \dfrac { 3^{14}-1 } {2} = 2\,391\,484 \\
\end{array}
$
}}$$

heureka Apr 29, 2015

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