Answer:
a) Neither
b) t(n) = n² - 8n + 16
Step-by-step explanation:
An Arithmetic Sequence has a constant difference between each consecutive term.
A Geometric Sequence has a constant ratio (multiplier) between each consecutive term.
Part (a)
As the sequence has neither a constant difference or a constant ratio, the sequence is neither arithmetic or geometric.
Part (b)
Work out the differences between the terms until the differences are the same:
First differences
[tex]16 \underset{-7}{\longrightarrow} 9 \underset{-5}{\longrightarrow} 4 \underset{-3}{\longrightarrow} 1 \underset{-1}{\longrightarrow} 0 \underset{+1}{\longrightarrow} 1 \underset{+3}{\longrightarrow} 4[/tex]
Second differences
[tex]-7 \underset{+2}{\longrightarrow} -5 \underset{+2}{\longrightarrow} -3\underset{+2}{\longrightarrow} -1\underset{+2}{\longrightarrow} 1\underset{+2}{\longrightarrow} 3[/tex]
As the second differences are the same, the sequence is quadratic and will contain an n² term. The coefficient of n² is always half of the second difference. Therefore, the coefficient of n² = 1.
Write out the numbers in the sequence n² and determine the operation that takes n² to the given sequence:
[tex]\begin{array}{|c|c|c|c|c|c|c|c|}\cline{1-8} n&0& 1& 2&3 &4 &5 &6 \\\cline{1-8}n^2 &0& 1& 4& 9&16 & 25&36 \\\cline{1-8} \sf operation&+16& +8&+0&-8&-16&-24&-32\\\cline{1-8} \sf sequence & 16&9 &4 & 1& 0& 1& 4\\\cline{1-8}\end{array}[/tex]
As the operation is not constant, work out the differences between the operations:
[tex]16\underset{-8}{\longrightarrow} 8\underset{-8}{\longrightarrow} 0\underset{-8}{\longrightarrow} -8\underset{-8}{\longrightarrow} -16\underset{-8}{\longrightarrow} -24\underset{-8}{\longrightarrow} -32[/tex]
As the differences are the same, the second operation in the sequence is -8n. Write out the numbers in the sequence with both operations and and determine the operation that takes (n² - 8n) to the given sequence:
[tex]\begin{array}{|c|c|c|c|c|c|c|c|}\cline{1-8} n&0& 1& 2&3 &4 &5 &6 \\\cline{1-8}n^2 -8n&-0&-7&-12&-15&-16&-15&-12\\\cline{1-8}\sf operation &+16&+16&+16&+16&+16&+16&+16\\\cline{1-8} \sf sequence & 16&9 &4 & 1& 0& 1& 4\\\cline{1-8}\end{array}[/tex]
As the operation is constant, the final operation in the sequence is +16.
So the equation for the nth term is:
[tex]\implies t(n)=n^2-8n+16[/tex]
