`sum_(i = 1)^6 (6i - 8i^3)` Find the sum using formulas for the sums of powers of integers.

Given: `sum_(i=1)^6 (6i-8i^3)`

`=sum_(i=1)^6 6i -sum_(i=1)^6 8i^3`

`=6sum_(i=1)^6 i- 8sum_(i=1)^6 i^3`

Use the Sums of Powers of Integers to find the sums.

`1+2+3+4+...+n=[n(n+1)]/2`

`1^3+2^3+3^3+4^3+...+n^3=[n^2(n+1)^2]/4`

`=6[6(6+1)]/2-8[6^2(6+1)^2]/4`

`=6(21)-8(441)`

`=-3402`

The sum is -3402.