Use the identities to evaluate the sums below: `sum_(i=1)^n(8i^3+2)/n^4*sum_(i=1)^n(12i^2-10)/n^3`


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Evaluate `sum_(i=1)^n (8i^3+2)/n^4 * sum_(i=1)^n(12i^2-10)/n^3` :

First we recognize that `n` is a constant, and `sumk(f(i))=ksumf(i)`

So we can rewrite as :


Then we note that `sum(f+g)=sumf+sumg` .


Again we can pull out the constants. Also, `sum_(i=1)^na=an` where `a` is a constant. Thus we get:


Now we use the summation rules `sum_(i=1)^ni^2=n^3/3+n^2/2+n/6` and `sum_(i=1)^ni^3=n^4/4+n^3/2+n^2/4` :




** If you really meant to multiply here, you can use the distributive property. If you meant to find the two sums then:


`sum_(i=1)^n(8i^3+2)/n^4=2+4/n+2/n^2+2/n^3` and `sum_(i=1)^n(12i^2-10)/n^3=4+6/n-8/n^2`


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