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Given a geometric sequence where the 5th term = 162, and the 8th term = -4374, determine the first three terms of the sequence.

I am unclear how to do this when I was not given the first term or the common ratio. Please help!!

 Jul 8, 2015

Best Answer 

 #1
avatar+26364 
+10

Given a geometric sequence where the 5th term = 162, and the 8th term = -4374, determine the first three terms of the sequence.

I am unclear how to do this when I was not given the first term or the common ratio. Please help!!

 

$$\begin{array}{rclr}
\small{ \text{geometric sequence: } a_n &=& a_1\cdot r^{n-1} }\\\\
\small{ \text{we have: } a_5 &=& a_1\cdot r^{5-1} = a_1 \cdot r^4 = & 162 }\\
\small{ \text{and we have: } a_8 &=& a_1\cdot r^{8-1} = a_1 \cdot r^7 = & -4374 }
\end{array}\\\\
\small{\text{If we divide }}
a_5 \small{\text{ and } a_8, \small{\text{we can calculate the ratio r }}$$

 

$$\small{\text{
$
\begin{array}{rclr}
\dfrac{ a_8 } { a_5 } &=& \dfrac{a_1\cdot r^7}{a_1 \cdot r^4} = \dfrac{-4374 }{ 162} \\\\
\dfrac{a_1\cdot r^7}{a_1 \cdot r^4} &=& \dfrac{-4374 }{ 162} \\\\
\dfrac{r^7}{r^4} &=& \dfrac{-4374 }{ 162} \\\\
r^{7-4} &=& \dfrac{-4374 }{ 162} \\\\
r^{3} &=& \dfrac{-4374 }{ 162} \\\\
r^{3} &=& -27 \qquad | \qquad \sqrt[3]{}\\\\
\mathbf{r} &\mathbf{=}&\mathbf{ -3 }
\end{array}
$}}$$

 

$$\small{\text{Now we can calculate the first term }} a_1
\small{\text{ with } a_5 \small{\text{ or } a_8$$

 

$$\small{\text{
$
\begin{array}{rclr}
a_5 = a_1 \cdot r^4 &=& 162 \\\\
a_1 \cdot (-3)^4 &=& 162 \\\\
a_1 \cdot 81 &=& 162 \qquad | \qquad :81\\\\
\mathbf{a_1} &\mathbf{=}& \mathbf{2} \\\\
\end{array}
$}}$$

 

$$\\\small{\text{check:}}\\
\small{\text{
$
\begin{array}{rclr}
a_8 &=& a_1 \cdot r^7 \\\\
a_8 &=& 2\cdot (-3)^7 \\\\
a_8 &=& 2\cdot( -2187 )\\\\
\mathbf{a_8} &\mathbf{=}& \mathbf{-4374} \qquad \text{ okay } \\\\
\end{array}
$}}$$

 

 Jul 8, 2015
 #1
avatar+26364 
+10
Best Answer

Given a geometric sequence where the 5th term = 162, and the 8th term = -4374, determine the first three terms of the sequence.

I am unclear how to do this when I was not given the first term or the common ratio. Please help!!

 

$$\begin{array}{rclr}
\small{ \text{geometric sequence: } a_n &=& a_1\cdot r^{n-1} }\\\\
\small{ \text{we have: } a_5 &=& a_1\cdot r^{5-1} = a_1 \cdot r^4 = & 162 }\\
\small{ \text{and we have: } a_8 &=& a_1\cdot r^{8-1} = a_1 \cdot r^7 = & -4374 }
\end{array}\\\\
\small{\text{If we divide }}
a_5 \small{\text{ and } a_8, \small{\text{we can calculate the ratio r }}$$

 

$$\small{\text{
$
\begin{array}{rclr}
\dfrac{ a_8 } { a_5 } &=& \dfrac{a_1\cdot r^7}{a_1 \cdot r^4} = \dfrac{-4374 }{ 162} \\\\
\dfrac{a_1\cdot r^7}{a_1 \cdot r^4} &=& \dfrac{-4374 }{ 162} \\\\
\dfrac{r^7}{r^4} &=& \dfrac{-4374 }{ 162} \\\\
r^{7-4} &=& \dfrac{-4374 }{ 162} \\\\
r^{3} &=& \dfrac{-4374 }{ 162} \\\\
r^{3} &=& -27 \qquad | \qquad \sqrt[3]{}\\\\
\mathbf{r} &\mathbf{=}&\mathbf{ -3 }
\end{array}
$}}$$

 

$$\small{\text{Now we can calculate the first term }} a_1
\small{\text{ with } a_5 \small{\text{ or } a_8$$

 

$$\small{\text{
$
\begin{array}{rclr}
a_5 = a_1 \cdot r^4 &=& 162 \\\\
a_1 \cdot (-3)^4 &=& 162 \\\\
a_1 \cdot 81 &=& 162 \qquad | \qquad :81\\\\
\mathbf{a_1} &\mathbf{=}& \mathbf{2} \\\\
\end{array}
$}}$$

 

$$\\\small{\text{check:}}\\
\small{\text{
$
\begin{array}{rclr}
a_8 &=& a_1 \cdot r^7 \\\\
a_8 &=& 2\cdot (-3)^7 \\\\
a_8 &=& 2\cdot( -2187 )\\\\
\mathbf{a_8} &\mathbf{=}& \mathbf{-4374} \qquad \text{ okay } \\\\
\end{array}
$}}$$

 

heureka Jul 8, 2015

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