Solve (n+1)(n/3-1/3)/n

$$\frac { ( n + 1 ) ( \frac { n } { 3 } - \frac { 1 } { 3 } ) } { n }$$

Evaluate

$$\frac{n}{3}-\frac{1}{3n}$$

Short Solution Steps

Since \frac{n}{3} and \frac{1}{3} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\left(n+1\right)\times \frac{n-1}{3}}{n}$$

Express \left(n+1\right)\times \frac{n-1}{3} as a single fraction.

$$\frac{\frac{\left(n+1\right)\left(n-1\right)}{3}}{n}$$

Express \frac{\frac{\left(n+1\right)\left(n-1\right)}{3}}{n} as a single fraction.

$$\frac{\left(n+1\right)\left(n-1\right)}{3n}$$

Consider \left(n+1\right)\left(n-1\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

$$\frac{n^{2}-1^{2}}{3n}$$

Calculate 1 to the power of 2 and get 1.

$$\frac{n^{2}-1}{3n}$$

Expand

$$\frac{n}{3}-\frac{1}{3n}$$

Short Solution Steps

Since \frac{n}{3} and \frac{1}{3} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\left(n+1\right)\times \frac{n-1}{3}}{n}$$

Express \left(n+1\right)\times \frac{n-1}{3} as a single fraction.

$$\frac{\frac{\left(n+1\right)\left(n-1\right)}{3}}{n}$$

Express \frac{\frac{\left(n+1\right)\left(n-1\right)}{3}}{n} as a single fraction.

$$\frac{\left(n+1\right)\left(n-1\right)}{3n}$$

Consider \left(n+1\right)\left(n-1\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

$$\frac{n^{2}-1^{2}}{3n}$$

Calculate 1 to the power of 2 and get 1.

$$\frac{n^{2}-1}{3n}$$