1. **Stating the problem:** We have a sequence of figures numbered 1, 2, 3 with respective numbers of squares 5, 12, and 21. 2. **Drawing figures 1 and 3:** - Figure 1 has 5 squares. - Figure 3 has 21 squares. 3. **Finding the increase in squares from figure 3 to figure 4:** - Given answer: 11 squares increase. 4. **Finding a formula for the number of squares in figure n:** - The formula given is $$F_n = 4n(n + 1)$$. 5. **Explanation of the formula:** - The number of squares grows quadratically with n. - The term $4n(n+1)$ suggests the figure is composed of $n$ rows and $n+1$ columns of 4 squares each. 6. **Verification with given data:** - For $n=1$: $$F_1 = 4 \times 1 \times (1+1) = 4 \times 1 \times 2 = 8$$ but given is 5, so the formula might need adjustment. - For $n=2$: $$F_2 = 4 \times 2 \times 3 = 24$$ but given is 12. 7. **Adjusting the formula:** - The given data points are 5, 12, 21 for $n=1,2,3$. - Let's try to find a quadratic formula $F_n = an^2 + bn + c$. 8. **Using the points:** - For $n=1$: $a + b + c = 5$ - For $n=2$: $4a + 2b + c = 12$ - For $n=3$: $9a + 3b + c = 21$ 9. **Solving the system:** - Subtract first from second: $3a + b = 7$ - Subtract second from third: $5a + b = 9$ - Subtract these two: $(5a + b) - (3a + b) = 9 - 7 \Rightarrow 2a = 2 \Rightarrow a = 1$ - Substitute $a=1$ into $3a + b = 7$: $3(1) + b = 7 \Rightarrow b = 4$ - Substitute $a=1$, $b=4$ into $a + b + c = 5$: $1 + 4 + c = 5 \Rightarrow c = 0$ 10. **Final formula:** $$F_n = n^2 + 4n$$ 11. **Check with $n=3$:** $$3^2 + 4 \times 3 = 9 + 12 = 21$$ which matches the given data. 12. **Answer to part b:** - Increase from figure 3 to 4: $$F_4 - F_3 = (4^2 + 4 \times 4) - (3^2 + 4 \times 3) = (16 + 16) - (9 + 12) = 32 - 21 = 11$$ squares. **Final answers:** - a) Drawings of figures 1 and 3 (not shown here). - b) Increase in squares from figure 3 to 4 is 11. - c) Formula for number of squares in figure n is $$F_n = n^2 + 4n$$. This formula was derived by fitting a quadratic to the given data points.