Triangular Number Pairs with Triangular Sum and Difference/Examples/T5, T6

Example of Triangular Number Pair with Triangular Sum and Difference

The triangular numbers $T_5$ and $T_6$ have sum and difference which are themselves both triangular:

\(\ds T_5\)

\(=\)

\(\ds \frac {5 \times 6} 2\)

$= 15$

\(\ds T_6\)

\(=\)

\(\ds \frac {6 \times 7} 2\)

$= 21$

\(\ds T_6 - T_5\)

\(=\)

\(\ds 21 - 15\)

\(\ds \)

\(=\)

\(\ds 6\)

\(\ds \)

\(=\)

\(\ds \dfrac {3 \times 4} 2\)

\(\ds \)

\(=\)

\(\ds T_3\)

\(\ds T_6 + T_5\)

\(=\)

\(\ds 21 + 15\)

\(\ds \)

\(=\)

\(\ds 36\)

\(\ds \)

\(=\)

\(\ds \dfrac {8 \times 9} 2\)

\(\ds \)

\(=\)

\(\ds T_8\)

$\blacksquare$