Answer
The statement $2+4+6++2n=n^2+n+2$ is not true by the Principal of Mathematical Induction.
Work Step by Step
The principle of mathematical induction states:
a) The statement must hold for the first natural number, known as the base case.
b) Assume that the statement is true for some arbitrary natural number $n$ larger than the first natural number. Then we will prove that the statement also holds for $n + 1$. This is known as the inductive step.
We need to prove that $2+4+6++2n=n^2+n+2$ is true for all natural numbers.
Let us check for $n=k$
So, we have:
$2+4+6++2k=k^2+k+2$
Let us check for $n=k+1$.
So, we have:
$2+4+6++2k+2(k+1)=k^2+k+2+2(k+1)\\=k^2+3k+4\\=k^2+2k+1+(k+1)+2\\=(k+1)^2+(k+1)+2$
Let us check for $n=1$. We have:
$2 \ne (1)^2+1+2\implies 2\ne 4$
So, $p(1)$ is not true.
Thus, the statement $2+4+6++2n=n^2+n+2$ is not true by the Principal of Mathematical Induction.
