Question

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

$n^2+n$is divisible by$2$.

Short answer

The statement is shown.

Step-by-step solution

Step 1. Given information

The expression is$n^2+n$.

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

Substitute $n=1$in $n^2+n$.

$(1{)}^2+1=2$which is divisible by $2$.

Thus, the statement is true for $n=1$

Assume that the statement is true for $n=k$

Thus, $k^2+k$is divisible by $2$

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

Since $k^2+k$is divisible by 2 and $2(k+1)$is also divisible by $2,(k+1{)}^2+(k+1)$is divisible by 2 .

Thus, the statement is true for $k+1$.

Since both the conditions of the Principle of Mathematical Induction are satisfied, the statement is true for all natural numbers.

Therefore, $n^2+n$is divisible by 2 .