Show how the 6th square number is formed from the sum of two consecutive triangular numbers.

The question asks about the $6^{th}$ square number.
Following the pattern shown in the examples:

• The $2^{nd}$ square number ($2^2=4$) is made from the $1^{st}$ and $2^{nd}$ triangular numbers ($T_1+T_2 = 1+3=4$).
• The $3^{rd}$ square number ($3^2=9$) is made from the $2^{nd}$ and $3^{rd}$ triangular numbers ($T_2+T_3 = 3+6=9$).
• The $4^{th}$ square number ($4^2=16$) is made from the $3^{rd}$ and $4^{th}$ triangular numbers ($T_3+T_4 = 6+10=16$).
• The $5^{th}$ square number ($5^2=25$) is made from the $4^{th}$ and $5^{th}$ triangular numbers ($T_4+T_5 = 10+15=25$).

Therefore, the $6^{th}$ square number ($6^2=36$) will be made from the $5^{th}$ triangular number ($T_5$) and the $6^{th}$ triangular number ($T_6$).

The formula for the $n^{th}$ triangular number is $T_n = \frac{n(n+1)}{2}$.
For the $5^{th}$ triangular number ($T_5$):

$$T_5 = \frac{5(5+1)}{2} = \frac{5 \times 6}{2} = \frac{30}{2} = 15$$

For the $6^{th}$ triangular number ($T_6$):

$$T_6 = \frac{6(6+1)}{2} = \frac{6 \times 7}{2} = \frac{42}{2} = 21$$

The sum of the $5^{th}$ and $6^{th}$ triangular numbers is:

$$T_5 + T_6 = 15 + 21 = 36$$

This is equal to $6^2$, the $6^{th}$ square number.

To draw the pattern for the $6^{th}$ square number (36 dots):

• Draw the $6^{th}$ triangular number ($T_6$) as a right-angled triangle of dots. This pattern will have 6 dots in the bottom row, 5 in the row above it, and so on, up to 1 dot at the top.
• Draw the $5^{th}$ triangular number ($T_5$) as another right-angled triangle of dots. This pattern will have 5 dots in the bottom row, 4 in the row above it, and so on, up to 1 dot at the top.
• Rotate the $T_5$ pattern by $180^\circ$ and place it adjacent to the $T_6$ pattern. The combined patterns will form a $6 \times 6$ square.

The $6^{th}$ square number is 36. It is made from the $5^{th}$ triangular number (15) and the $6^{th}$ triangular number (21).