Solve n(n+1)+2(n+1)/2=(n+1)(n+2)/2

$$\frac { n ( n + 1 ) + 2 ( n + 1 ) } { 2 } = \frac { ( n + 1 ) ( n + 2 ) } { 2 }$$

Solve for n (complex solution)

$$n\in \mathrm{C}$$

Solution Steps

Multiply both sides of the equation by 2.

$$n\left(n+1\right)+2\left(n+1\right)=\left(n+1\right)\left(n+2\right)$$

Use the distributive property to multiply n by n+1.

$$n^{2}+n+2\left(n+1\right)=\left(n+1\right)\left(n+2\right)$$

Use the distributive property to multiply 2 by n+1.

$$n^{2}+n+2n+2=\left(n+1\right)\left(n+2\right)$$

Combine n and 2n to get 3n.

$$n^{2}+3n+2=\left(n+1\right)\left(n+2\right)$$

Use the distributive property to multiply n+1 by n+2 and combine like terms.

$$n^{2}+3n+2=n^{2}+3n+2$$

Subtract n^{2} from both sides.

$$n^{2}+3n+2-n^{2}=3n+2$$

Combine n^{2} and -n^{2} to get 0.

$$3n+2=3n+2$$

Subtract 3n from both sides.

$$3n+2-3n=2$$

Combine 3n and -3n to get 0.

$$2=2$$

Compare 2 and 2.

$$\text{true}$$

This is true for any n.

$$n\in \mathrm{C}$$

Solve for n

$$n\in \mathrm{R}$$

Solution Steps

Multiply both sides of the equation by 2.

$$n\left(n+1\right)+2\left(n+1\right)=\left(n+1\right)\left(n+2\right)$$

Use the distributive property to multiply n by n+1.

$$n^{2}+n+2\left(n+1\right)=\left(n+1\right)\left(n+2\right)$$

Use the distributive property to multiply 2 by n+1.

$$n^{2}+n+2n+2=\left(n+1\right)\left(n+2\right)$$

Combine n and 2n to get 3n.

$$n^{2}+3n+2=\left(n+1\right)\left(n+2\right)$$

Use the distributive property to multiply n+1 by n+2 and combine like terms.

$$n^{2}+3n+2=n^{2}+3n+2$$

Subtract n^{2} from both sides.

$$n^{2}+3n+2-n^{2}=3n+2$$

Combine n^{2} and -n^{2} to get 0.

$$3n+2=3n+2$$

Subtract 3n from both sides.

$$3n+2-3n=2$$

Combine 3n and -3n to get 0.

$$2=2$$

Compare 2 and 2.

$$\text{true}$$

This is true for any n.

$$n\in \mathrm{R}$$