Discussion
If \(u_{1} = 1\), \(u_{3} = 17\), \(u_{4} = 43\) and \(u_{5} = 89\), then value of \(u_{2}\) is :
Answer: B
\(u_{2} = \) 5.
With four known values the missing one is found by assuming the data fit a cubic, so the fourth difference over five consecutive values is 0:
\[ \begin{aligned} \Delta^{4}u_{1} &= u_{5} - 4u_{4} + 6u_{3} - 4u_{2} + u_{1} = 0 \\ 89 - 172 + 102 - 4u_{2} + 1 &= 0 \\ 20 &= 4u_{2},\ u_{2} = 5 \end{aligned} \]
Check: the values 1, 5, 17, 43, 89 have differences 4, 12, 26, 46, then 8, 14, 20, then 6, 6, constant third differences as a cubic requires. The other options break that pattern.
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