Show, using the formulas for sums of fourth powers and...

Show, using the formulas for sums of fourth powers and squares, that
$$
\begin{aligned}
\sum_{i=1}^{n-1}\left(n^{4}-2 n^{2} i^{2}+i^{4}\right) &=\frac{8}{15}(n-1) n^{4}+\frac{1}{30} n^{4}-\frac{1}{30} n \\
&=\frac{8}{15} n \cdot n^{4}-\frac{1}{2} n^{4}-\frac{1}{30} n
\end{aligned}
$$

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Key Concepts

Summation Formulas

These are formulas that provide closed-form expressions for the sum of powers of integers, such as the sum of squares or the sum of fourth powers. They are essential in converting a summation expression into a polynomial form which can be simplified and manipulated algebraically.

Sum of Squares

This concept involves finding a closed-form expression for the sum of the squares of the first n integers. The formula allows one to replace a summation notation with its equivalent polynomial expression, which simplifies the analysis of problems involving quadratic terms.

Sum of Fourth Powers

Similar to the sum of squares, this concept provides a formula for the sum of the fourth powers of the first n integers. This formula is critical when evaluating expressions containing quartic terms, enabling the transformation of a summation into a simplified polynomial form.

Algebraic Manipulation of Polynomials

This concept involves techniques for expanding, factoring, and simplifying polynomial expressions. By using algebraic manipulation, one can verify identities and reduce complex summations into simpler polynomial forms, which is often necessary when confirming equality between different expressions.

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