+0  
 
0
79
1
avatar

whats the nth term of 2,8,18,32

Guest Feb 22, 2017

Best Answer 

 #1
avatar+18612 
+25

whats the nth term of 2,8,18,32

 

\(\small{ \begin{array}{lrrrrrrrrrr} & {\color{red}d_0 = 2} && 8 && 18 && 32 && 50 && \cdots \\ \text{1. Difference } && {\color{red}d_1 = 6} && 10 && 14 && 18 && \cdots \\ \text{2. Difference } &&& {\color{red}d_2 = 4} && 4 && 4 && \cdots \\ \end{array} }\)

 

 

\(\boxed{~ \begin{array}{rcl} a_n &=& \dbinom{n-1}{0}\cdot {\color{red}d_0 } + \dbinom{n-1}{1}\cdot {\color{red}d_1 } + \dbinom{n-1}{2}\cdot {\color{red}d_2 } \end{array} ~}\)

 

\(\begin{array}{|rcll|} \hline a_n &=& \dbinom{n-1}{0}\cdot {\color{red} 2 } + \dbinom{n-1}{1}\cdot {\color{red} 6 } + \dbinom{n-1}{2}\cdot {\color{red} 4 } \\\\ && \binom{n-1}{0} = 1 \\ && \binom{n-1}{1} = n-1 \\ && \binom{n-1}{2} = \left( \frac{n-1}{2} \right) \cdot \left( \frac{n-2}{1} \right)\\\\ a_n &=& {\color{red} 2 } + (n-1)\cdot {\color{red} 6 } + \left( \frac{n-1}{2} \right) \cdot (n-2) \cdot {\color{red} 4 } \\ &=& {\color{red} 2 } + (n-1)\cdot {\color{red} 6 } + (n-1) \cdot (n-2) \cdot {\color{red} 2 } \\ &=& 2 + 6n-6 + (2n-2) \cdot (n-2) \\ &=& -4 + 6n + 2n^2-4n-2n+4 \\ \mathbf{a_n} & \mathbf{=} & \mathbf{2n^2} \\ \hline \end{array}\)

 

Example:

\(\begin{array}{|rclcl|} \hline a_1 &=& 2\cdot 1^2 &=& 2 \\ a_2 &=& 2\cdot 2^2 &=& 8 \\ a_3 &=& 2\cdot 3^2 &=& 18 \\ a_4 &=& 2\cdot 4^2 &=& 32 \\ a_5 &=& 2\cdot 5^2 &=& 50 \\ \dots \\ \mathbf{a_n} &\mathbf{=}& \mathbf{2\cdot n^2} \\ \hline \end{array}\)

 

laugh

heureka  Feb 22, 2017
Sort: 

1+0 Answers

 #1
avatar+18612 
+25
Best Answer

whats the nth term of 2,8,18,32

 

\(\small{ \begin{array}{lrrrrrrrrrr} & {\color{red}d_0 = 2} && 8 && 18 && 32 && 50 && \cdots \\ \text{1. Difference } && {\color{red}d_1 = 6} && 10 && 14 && 18 && \cdots \\ \text{2. Difference } &&& {\color{red}d_2 = 4} && 4 && 4 && \cdots \\ \end{array} }\)

 

 

\(\boxed{~ \begin{array}{rcl} a_n &=& \dbinom{n-1}{0}\cdot {\color{red}d_0 } + \dbinom{n-1}{1}\cdot {\color{red}d_1 } + \dbinom{n-1}{2}\cdot {\color{red}d_2 } \end{array} ~}\)

 

\(\begin{array}{|rcll|} \hline a_n &=& \dbinom{n-1}{0}\cdot {\color{red} 2 } + \dbinom{n-1}{1}\cdot {\color{red} 6 } + \dbinom{n-1}{2}\cdot {\color{red} 4 } \\\\ && \binom{n-1}{0} = 1 \\ && \binom{n-1}{1} = n-1 \\ && \binom{n-1}{2} = \left( \frac{n-1}{2} \right) \cdot \left( \frac{n-2}{1} \right)\\\\ a_n &=& {\color{red} 2 } + (n-1)\cdot {\color{red} 6 } + \left( \frac{n-1}{2} \right) \cdot (n-2) \cdot {\color{red} 4 } \\ &=& {\color{red} 2 } + (n-1)\cdot {\color{red} 6 } + (n-1) \cdot (n-2) \cdot {\color{red} 2 } \\ &=& 2 + 6n-6 + (2n-2) \cdot (n-2) \\ &=& -4 + 6n + 2n^2-4n-2n+4 \\ \mathbf{a_n} & \mathbf{=} & \mathbf{2n^2} \\ \hline \end{array}\)

 

Example:

\(\begin{array}{|rclcl|} \hline a_1 &=& 2\cdot 1^2 &=& 2 \\ a_2 &=& 2\cdot 2^2 &=& 8 \\ a_3 &=& 2\cdot 3^2 &=& 18 \\ a_4 &=& 2\cdot 4^2 &=& 32 \\ a_5 &=& 2\cdot 5^2 &=& 50 \\ \dots \\ \mathbf{a_n} &\mathbf{=}& \mathbf{2\cdot n^2} \\ \hline \end{array}\)

 

laugh

heureka  Feb 22, 2017

22 Online Users

avatar
avatar
avatar
avatar
We use cookies to personalise content and ads, to provide social media features and to analyse our traffic. We also share information about your use of our site with our social media, advertising and analytics partners.  See details