Growing figures

Identical matchsticks are laid to make a row of squares, the same way as before: figure 1 uses 4 matchsticks, figure 2 uses 7 and figure 3 uses 10. One figure in this pattern uses 61 matchsticks. Which figure is it? Build a table beside the figures. Figure 1 is 1 square, and it uses 4 matchsticks. In figure 2 the 2 squares share a matchstick. That shared one is counted once, so figure 2 uses 7. Figure 3 uses 10. Look at the difference between 2 rows. 7 minus 4 is 3. 10 minus 7 is 3 as well. The same difference twice is what lets us name a rule. The count goes up by 3 each time. Figure 1 is the starting row, and nothing has been added to it yet. Reach for figure 12. Every arrow crosses one gap, and from figure 1 to figure 12 there are 11 gaps. 12 minus 1 is 11. Now read the table the other way, from a count back to a figure. Figure 1 already holds 4, so 61 minus 4 is 57 matchsticks added after it. Each gap adds 3, so 57 divided by 3 is 19 gaps. 19 gaps after figure 1 lands on figure 20.

Your turn. Figure 1 is a cross made of 5 identical squares and is 12 centimetres tall. Figure 2 is made by adding one square to the end of each of the four arms of figure 1. Every figure after that is made from the one before it in the same way. Find the area of figure 25. Try it first. Pause if you like. The only length given is figure 1's height. The cross is 3 squares tall, so 12 divided by 3 is 4 centimetres for one square's side. 4 times 4 is 16 square centimetres for one square. Now count squares, and build the same table. Figure 1 has 5, figure 2 has 9, figure 3 has 13. 9 minus 5 is 4, and 13 minus 9 is 4 too. From figure 1 to figure 25 there are 24 gaps. 5 plus 24 times 4 is one hundred and one squares. A count is not an area, so they stay apart until the end. one hundred and one squares, each 16 square centimetres, is one thousand, six hundred and sixteen square centimetres. A figure number counts the rows. Every gap between them adds the same difference. A growing pattern is a first row and the same difference repeated.