Triangular Number Pairs with Triangular Sum and Difference/Examples
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Examples of Triangular Number Pairs with Triangular Sum and Difference
$T_5$ and $T_6$
The triangular numbers $T_5$ and $T_6$ have sum and difference which are themselves both triangular:
\(\ds T_5\) | \(=\) | \(\ds \frac {5 \times 6} 2\) | $= 15$ | |||||||||||
\(\ds T_6\) | \(=\) | \(\ds \frac {6 \times 7} 2\) | $= 21$ |
\(\ds T_6 - T_5\) | \(=\) | \(\ds 21 - 15\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 6\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac {3 \times 4} 2\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds T_3\) |
\(\ds T_6 + T_5\) | \(=\) | \(\ds 21 + 15\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 36\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac {8 \times 9} 2\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds T_8\) |
$\blacksquare$
$T_{14}$ and $T_{18}$
The triangular numbers $T_{14}$ and $T_{18}$ have sum and difference which are themselves both triangular:
\(\ds T_{14}\) | \(=\) | \(\ds \frac {14 \times 15} 2\) | $= 105$ | |||||||||||
\(\ds T_{18}\) | \(=\) | \(\ds \frac {18 \times 19} 2\) | $= 171$ |
\(\ds T_{18} - T_{14}\) | \(=\) | \(\ds 171 - 105\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 66\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac {11 \times 12} 2\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds T_{11}\) |
\(\ds T_{18} + T_{14}\) | \(=\) | \(\ds 171 + 105\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 276\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac {23 \times 24} 2\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds T_{23}\) |
$\blacksquare$
$T_{27}$ and $T_{37}$
The triangular numbers $T_{27}$ and $T_{37}$ have sum and difference which are themselves both triangular:
\(\ds T_{27}\) | \(=\) | \(\ds \frac {27 \times 28} 2\) | $= 378$ | |||||||||||
\(\ds T_{37}\) | \(=\) | \(\ds \frac {37 \times 38} 2\) | $= 703$ |
\(\ds T_{37} - T_{27}\) | \(=\) | \(\ds 703 - 378\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 325\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \frac {25 \times 26} 2\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds T_{25}\) |
\(\ds T_{37} + T_{27}\) | \(=\) | \(\ds 703 + 378\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 1081\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \frac {46 \times 47} 2\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds T_{46}\) |
$\blacksquare$
$T_{39}$ and $T_{44}$
The triangular numbers $T_{39}$ and $T_{44}$ have sum and difference which are themselves both triangular:
\(\ds T_{39}\) | \(=\) | \(\ds \frac {39 \times 40} 2\) | $= 780$ | |||||||||||
\(\ds T_{44}\) | \(=\) | \(\ds \frac {44 \times 45} 2\) | $= 990$ |
\(\ds T_{44} - T_{39}\) | \(=\) | \(\ds 990 - 780\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 210\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac {20 \times 21} 2\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds T_{20}\) |
\(\ds T_{44} + T_{39}\) | \(=\) | \(\ds 990 + 780\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 1770\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac {59 \times 60} 2\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds T_{59}\) |
$\blacksquare$
$T_{1869}$ and $T_{2090}$
The triangular numbers $T_{1869}$ and $T_{2090}$ have sum and difference which are themselves both triangular:
\(\ds T_{1869}\) | \(=\) | \(\ds \frac {1869 \times 1870} 2\) | \(\ds = 1 \, 747 \, 515\) | |||||||||||
\(\ds T_{2090}\) | \(=\) | \(\ds \frac {2090 \times 2091} 2\) | \(\ds = 2 \, 185 \, 095\) |
\(\ds T_{2090} - T_{1869}\) | \(=\) | \(\ds 2 \, 185 \, 095 - 1 \, 747 \, 515\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 437 \, 580\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac {935 \times 936} 2\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds T_{935}\) |
\(\ds T_{2090} + T_{1869}\) | \(=\) | \(\ds 2 \, 185 \, 095 - 1 \, 747 \, 515\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 3 \, 932 \, 610\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac {2804 \times 2805} 2\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds T_{2804}\) |
$\blacksquare$