Question

Let d ∈ Z+ be a positive integer. Prove that 1(d) + 2(d) + · · · + n(d) ∈ O(n(d+1)). Clearly state what your witnesses are.

Answer 1

To prove that 1(d) + 2(d) + · · · + n(d) ∈ O(n(d+1)), the sum formula for an arithmetic series and the binomial theorem can be used.

Step-by-step explanation:

To prove that 1(d) + 2(d) + · · · + n(d) ∈ O(n(d+1)), we can use the sum formula for an arithmetic series and the binomial theorem.

The sum of an arithmetic series is given by the formula Sn = (n/2)(a + l), where n is the number of terms, a is the first term, and l is the last term. In this case, a = 1(d) and l = n(d).

Using the binomial theorem, we can see that n(d) = n(d + 1) - nd. Therefore, the sum Sn = (n/2)(1(d) + n(d + 1)), which is equivalent to O(n(d + 1)).