Question

In Problem, use the Principle of Mathematical Induction to show that the given statement is true for all natural numbers $n$.

$-2-3-4-...-(n+1)=-\frac{1}{2}n(n+3)$

Short answer

The statement is shown for all the natural numbers.

Step-by-step solution

Step 1. Given information

The statement is$-2-3-4-...-(n+1)=-\frac{1}{2}n(n+3)$

Step 2. Show that the given statement is true for all natural numbers.

Put $n=1$in the given statement.

$$\begin{gathered} -2==-\frac{1}{2}\left(1\right)(1+3) \\ -2=-2 \end{gathered}$$

Next, we assume that the above statement is true for some $k$. so that localid="1647674105542" $-2-3-4\cdots -(k+1)=-\frac{1}{2}k(k+3)$

For width="61">n=k+1

$\begin{array}{rcl}-\frac{1}{2}k(k+3)-(k+2)& =& =-\left(\frac{k^2+3k+2(k+2)}{2}\right)\\ & =& -\left(\frac{k^2+3k+2k+4}{2}\right)\end{array}$