rove by induction that for all n ∈ N,5 + 7 + . . . + (2n + 3)= ∑i=ni=1 (2i + 3) = n(n + 4)

Sure, let's prove this by induction.

Step 1: Base Case Let's start with n=1. The left side of the equation becomes 21 + 3 = 5. The right side of the equation becomes 1(1+4) = 5. Since both sides are equal, the base case holds true.

Step 2: Inductive Step Assume the statement is true for some k ∈ N, i.e., 5 + 7 + ... + (2k + 3) = k*(k + 4).

Step 3: Inductive Hypothesis We need to prove that the statement is true for k+1, i.e., 5 + 7 + ... + (2k + 3) + (2(k+1) + 3) = (k+1)*((k+1) + 4).

Substitute the inductive hypothesis into the left side of the equation:

5 + 7 + ... + (2k + 3) + (2(k+1) + 3) = k*(k + 4) + (2(k+1) + 3) = k^2 + 4k + 2k + 5 = k^2 + 6k + 5.

The right side of the equation is (k+1)((k+1) + 4) = (k+1)(k+5) = k^2 + 6k + 5.

Since both sides of the equation are equal, the statement is true for k+1 if it's true for k.

Therefore, by the principle of mathematical induction, the statement is true for all n ∈ N.