How do I get the equation for the 10th, 20th and nth members in the sequence? The sequence is, 1. 1x1; 1 square; 4 matchsticks 2. 2x2; 2 squares; 12 matchsticks 3. 3x3; 9 squares; 24 matchsticks   How do I find the pattern to make the equation to find the 10th, 20th, and nth members of the sequence?

Expert Answers

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10x10 square

Each horizontal lines will have 10 matchsticks.

You will have 11 horizontal lines

Therefore you will have 10*11 horizontal matchsticks

Same work with the vertical matchsticks

You will have 11 vertical lines of 10 matchsticks.

therefore you will have 11*10=110 vertical matchsticks.


In total you will have 220=2*11*10


If you have a 20x20 square. 

Again you will have 21 horizontal lines of 20 matchsticks and 21 vertical lines of 20 matchsticks. Therefore you have 20*21*2 matchsticks in total.


Now for a nxn square

we have horizontal n+1 horizontal lines of n matchsticks and n+1 vertical lines  of n matchsticks. In total we have n*(n+1)*2 matchsticks.


Therefore the formula is 2n(n+1) matchsticks for a n x n square.

It is matches the result for n=10, n=20 and of course n=1 n=2, n=3.


Formula: 2n(n+1)



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