"Counting in their heads" - 1895 oil painting

[Gerson W. Barbosa]

Alternatively, you can use this formula:

n[c + (n² - 1)/12]

where n is the number of elements in the sequence and c is the central element.

5[12² + (5² - 1)/12] = 5[144 + 2] = 730

This should work also for sequences with even number of elements. For instance,

5² + 6² + 7² + 8² = 4[6.5² + (4² - 1)/12] = 174

A bit more difficult to do it in one’s head, though.