# 7140

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## Number

$7140$ (seven thousand, one hundred and forty) is:

$2^2 \times 3 \times 5 \times 7 \times 17$

The $5$th and last after $1$, $10$, $120$, $1540$ of the $5$ tetrahedral numbers which are also triangular.

The $34$th tetrahedral number, after $1$, $4$, $10$, $20$, $35$, $\ldots$, $3276$, $3276$, $4060$, $4495$, $4960$, $5456$, $5984$, $6545$:
$7140 = \ds \sum_{k \mathop = 1}^{34} \frac {k \paren {k + 1} } 2 = \dfrac {34 \paren {34 + 1} \paren {34 + 2} } 6$

The $60$th hexagonal number after $1$, $6$, $15$, $28$, $45$, $66$, $91$, $\ldots$, $5565$, $5778$, $5995$, $6216$, $6441$, $6670$, $6903$:
$7140 = \ds \sum_{k \mathop = 1}^{60} \paren {4 k - 3} = 60 \paren {2 \times 60 - 1}$

The $119$th triangular number after $1$, $3$, $6$, $10$, $15$, $\ldots$, $5995$, $6105$, $6216$, $6328$, $6441$, $6555$, $6670$, $6786$, $6903$, $7021$:
$7140 = \ds \sum_{k \mathop = 1}^{119} k = \dfrac {119 \times \paren {119 + 1} } 2$