Question

PLEASE HELP BEEN STUCK ON THIS FORVER! AND ITS DUE SOON! WILL GET 100 POINTS FOR HELPING CORRECTLY!

FIND THE 100TH LINE SEGMENT
CUBE #1 HAS 4 LINES
CUBE #2 HAS 12
CUBE #3 HAS 24
WHATS THE 100TH LINE SEGMENT USING NTH TERM.
PICTURE BELOW!

Answer 1

20,200

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:

`4 \underset{+8}{\longrightarrow} 12 \underset{+12}{\longrightarrow} 24 \underset{+16}{\longrightarrow} 40`

`8 \underset{+4}{\longrightarrow} 12 \underset{+4}{\longrightarrow} 16`

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.

`\begin{array}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}`

From inspection of the table, we can see that the "operation" is to add 2n to 2n².

Therefore, the nth term is:

`a_n=2n^2+2n`

To find the number of lines in the 100th cube, substitute n = 100 into the equation for the nth term:

`\begin{aligned}n=100 \implies a_(100) & = 2(100)^2+2(100)\\& = 2(10000)+200)\\& = 20000+200\\& = 20200\end{aligned}`

Therefore, the 100th cube has 20,200 lines.