# 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?

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)

matchsticks

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