Answer:
20,200
Step-by-step explanation:
Let the number of lines in each cube be a term in the sequence:
4, 12, 24, 40, ...
Work out the differences between the terms until the differences are the same:
[tex]4 \underset{+8}{\longrightarrow} 12 \underset{+12}{\longrightarrow} 24 \underset{+16}{\longrightarrow} 40[/tex]
[tex]8 \underset{+4}{\longrightarrow} 12 \underset{+4}{\longrightarrow} 16[/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 is 2.
To work out the nth term of the sequence, write out the numbers in the sequence 2n² and compare this sequence with the given sequence.
[tex]\begin{array}{|c|c|c|c|c|}\cline{1-5} n & 1 & 2 & 3 & 4\\\cline{1-5}2n^2 & 2 & 8 & 18 & 32\\\cline{1-5}\sf operation & +2&+4&+6&+8 \\\cline{1-5}\sf sequence & 4 & 12 & 24 & 40\\\cline{1-5}\end{array}[/tex]
From inspection of the table, we can see that the "operation" is to add 2n to 2n².
Therefore, the nth term is:
[tex]a_n=2n^2+2n[/tex]
To find the number of lines in the 100th cube, substitute n = 100 into the equation for the nth term:
[tex]\begin{aligned}n=100 \implies a_{100} & = 2(100)^2+2(100)\\& = 2(10000)+200)\\& = 20000+200\\& = 20200\end{aligned}[/tex]
Therefore, the 100th cube has 20,200 lines.
