Which of the following expressions represents an even number for all integer values of $\newline$ $n$ ? $\newline$ $n+2$ , $\newline$ $2n$ , $\newline$ $2n-1$ , $\newline$ $n^{2}$ , $\newline$ $2n+1$

Full solution

Q. Which of the following expressions represents an even number for all integer values of

\newline

n

\newline

n+2

\newline

2n

\newline

2n-1

\newline

n^{2}

\newline

2n+1

Expression $1$ Analysis: Let's analyze each expression one by one to determine if it represents an even number for all integer values of $n$ . $\newline$ Expression $1$ : $n + 2$ $\newline$ If $n$ is an integer, adding $2$ to it will not change its parity (odd or even status). If $n$ is even, $n + 2$ is even. If $n$ is odd, $n + 2$ is also even. Therefore, this expression represents an even number for all integer values of $n$ .
Expression $2$ Analysis: Expression $2$ : $2n$ Multiplying any integer $n$ by $2$ will always result in an even number, since even numbers are defined as multiples of $2$ . Therefore, this expression represents an even number for all integer values of $n$ .
Expression $3$ Analysis: Expression $3$ : $2n - 1$ Subtracting $1$ from an even number ( $2n$ ) will always result in an odd number. Therefore, this expression does not represent an even number for all integer values of $n$ .
Expression $4$ Analysis: Expression $4$ : $n^2$ $\newline$ The square of an integer $n$ can be either even or odd. If $n$ is even, $n^2$ is even. If $n$ is odd, $n^2$ is odd. Therefore, this expression does not represent an even number for all integer values of $n$ .
Expression $5$ Analysis: Expression $5$ : $2n + 1$ $\newline$ Adding $1$ to an even number ( $2n$ ) will always result in an odd number. Therefore, this expression does not represent an even number for all integer values of $n$ .