Question

Let $P\left(k\right):2+4+6+...+2k=k(k+1)+2$, then the statement $P(m+1)$ will be true if

• A. $P\left(1\right)$ is true
• B. $P\left(2\right)$ is true
• C. $P\left(m\right)$ is true
• D. none of these

Answer 1

Answer: (C)

$P\left(m\right)$ is true

Given $P\left(k\right):2+4+6+...+2k=k(k+1)+2$
We have $P(m+1)$ is true.
$\Rightarrow 2+4+6+.....+2m+2(m+1)=(m+1)(m+2)+2$
$\Rightarrow 2+4+6+.....+2m=(m+1)(m+2)+2-2(m+1)$
$\Rightarrow 2+4+6+.....+2m=(m+1)m+2$
Hence, $P\left(m\right)$ is true.
Also, $P\left(1\right)$ is not true as
$LHS=2$
$RHS=2+2=4$
Also, $P\left(2\right)$ is not true as
$LHS=2+4=6$
$RHS=2\left(3\right)+2=8$