Exercise 3 Use a suitable telescoping sum to find a simpler

Exercise 3. Use a suitable telescoping sum to find a simpler expression for the sum 1^4 + 2^4 + . . . + n^4 , where n belongs to N. Check you: answer by verifying that it works for small values of it. (Given that the answer is a polynomial of degree five in n, it suffices to check six different values of n to be sure that your answer is right. The reason is that if two polynomials of degree five agree at six different points, then these six points are roots of the difference of the two polynomials, and this difference is a polynomial of degree at most five. A polynomial of degree at most five can have at most five roots, unless it is the zero polynomial.)

Solution

Sum of powers of fours of first n natural numbers

1⁴+2⁴+3⁴+4⁴+