Final answer:
The student's question to find a closed formula for H(n) can be addressed using the binomial theorem and involves understanding the derivative of a binomial expansion.
Step-by-step explanation:
The student is asking to find a closed formula for the sum given by the function H(n)=∑k≥1k(nk)4^k. To solve this, we can utilize the binomial theorem, which is a cornerstone in combinatorics and allows for the expansion of expressions in the form (a + b)^n. More specifically, we need to recognize that the sum presented in the question can be represented as a derivative of a binomial expansion.
To find a closed form for H(n), we note a similarity with the derivative of the binomial expansion. If we have the expression (1 + 4x)^n, the n-th derivative of this function evaluated at x=0 will give us n! times the coefficient of x^n in the expansion. By differentiating repeatedly and multiplying by k, we can align this process with the sum given in H(n). This is because the term k(nk)4^k resembles the k-th term in the derivative of the binomial expansion.
Therefore, by carefully differentiating and manipulating the binomial expansion of the form (1 + 4x)^n and evaluating at x=0, we can derive the closed formula for the sum in H(n).