Fill the series 5, 7, 17, 47, 115, ? find the value in the "?".
\(\small{ \begin{array}{lrrrrrrrrrr} & {\color{red}d_0 = 5} && 7 && 17 && 47 && 115 && \cdots \\ \text{1. Difference } && {\color{red}d_1 = 2} && 10 && 30 && 68 && \cdots \\ \text{2. Difference } &&& {\color{red}d_2 = 8} && 20 && 38 && \cdots \\ \text{3. Difference } &&&& {\color{red}d_3 = 12} && 18 && \cdots \\ \text{4. Difference } &&&&& {\color{red}d_4 = 6} && \cdots \\ \end{array} } \)
\(\boxed{~ \begin{array}{rcl} a_n &=& \binom{n-1}{0}\cdot {\color{red}d_0 } + \binom{n-1}{1}\cdot {\color{red}d_1 } + \binom{n-1}{2}\cdot {\color{red}d_2 } + \binom{n-1}{3}\cdot {\color{red}d_3 } + \binom{n-1}{4}\cdot {\color{red}d_4 } \end{array} ~} \\\\ \)
\(\begin{array}{rcl} a_n &=& \binom{n-1}{0}\cdot {\color{red} 5 } + \binom{n-1}{1}\cdot {\color{red} 2 } + \binom{n-1}{2}\cdot {\color{red} 8 } + \binom{n-1}{3}\cdot {\color{red} 12 } + \binom{n-1}{4}\cdot {\color{red} 6 } \\\\ a_{6} &=& \binom{5}{0}\cdot {\color{red} 5 } + \binom{5}{1}\cdot {\color{red} 2 } + \binom{5}{2}\cdot {\color{red} 8 } + \binom{5}{3}\cdot {\color{red} 12 } + \binom{5}{4}\cdot {\color{red} 6 } \\\\ a_{6} &=& 1\cdot {\color{red} 5 } + 5\cdot {\color{red} 2 } + 10\cdot {\color{red} 8 } + 10\cdot {\color{red} 12 } + 5\cdot {\color{red} 6 } \\\\ a_{6} &=& 5+10+80+120+30 \\\\ \mathbf{a_{6}} & \mathbf{=}& \mathbf{245} \end{array}\)
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