Question

NO LINKS!!

a. Identify each sequence as arithmetic, geometric, or neither
b. If it is arithmetic or geometric, describe the sequence generator

n t(n)
0 16
1 9
2 4
3 1
4 0
5 1
6 4

Answer 1

a) Neither

b) t(n) = n² - 8n + 16

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

`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`

Second differences

`-7 \underset{+2}{\longrightarrow} -5 \underset{+2}{\longrightarrow} -3\underset{+2}{\longrightarrow} -1\underset{+2}{\longrightarrow} 1\underset{+2}{\longrightarrow} 3`

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:

`\begin{array}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}`

As the operation is not constant, work out the differences between the operations:

`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`

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:

`\begin{array}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}`

As the operation is constant, the final operation in the sequence is +16.

So the equation for the nth term is:

`\implies t(n)=n^2-8n+16`

Answer 2

• a) Neither,
• b) t(n) = (n - 4)²

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We observe that, the sequence is symmetric and all the terms are perfect squares:

• 16, 9, 4, 1, 0, 1, 4 ⇒ 4², 3², 2², 1², 0², 1², 2²

This is neither arithmetic nor geometric.

The zero term is 16 and the fourth term is 0 so the nth term would be:

• t(n) = (n - 4)²