The sum in question is
5 + 10 + 15 + … + 75 + 80 + 85
= 5 (1 + 2 + 3 + … + 15 + 16 + 17)
Now,
[tex]S = 1 + 2 + 3 + \cdots + 15 + 16 + 17[/tex]
has 17 terms. Reversing the order, we have
[tex]S = 17 + 16 + 15 + \cdots + 3 + 2 + 1[/tex]
Then
[tex]2S = (1+17) + (2+16) + (3+15) + \cdots + (17 + 1) = 17\cdot18 \implies S=17\cdot9 = 153[/tex]
Then the value of the sum we want is 5 times this, and 5•153 = 765.
