<s>[INST] Verify that $4r^3+r=\left(r+\frac{1}{2}\right)^4-\left(r-\frac{1}{2}\right)^4$ Must I verify that Left hand side(LHS)= Right hand side(RHS)or can I prove that RHS= LHS?
I don’t know how to prove from LHS=RHS. How to separate the $4r^3+r$ into two terms, i.e. $\displaystyle\Bigl(r+\frac{1}{2}\Bigr)^4-\Bigl(r-\frac{1}{2}\Bigr)^4$
After verifying, I will need to find $$\sum_{r=1}^n (4r^3+r)$$
This is a summation of finite series question.
 [/INST] $(x+y)^{4}-(x-y)^{3}=8x^{3}y+8xy^3$
Just substitute $x=r$ and $y=\frac{1}{2}$ to obtain
$4r^{3}+r=(r+\frac{1}{2})^{4}-(r-\frac{1}{2})^{4}$
As for the infinite sum,
$\sum_{r=1}^{n}({4r^{3}+r})=-\left( \frac{1}{2} \right)^4+\left( \frac{3}{2} \right)^4 - \left( \frac{3}{2} \right)^4+...+\left( n + \frac{1}{2} \right)^4$
$\sum_{r=1}^{n}(4r^{3}+r)=\left( n+\frac{1}{2} \right)^4-\left( \frac{1}{2} \right)^4$
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