Solve d/2101^2+left({a}_{1}-d/2right)(101)=2020

$$\frac{ d }{ 2 } { 101 }^{ 2 } + \left( { a }_{ 1 } - \frac{ d }{ 2 } \right) (101) = 2020$$

Solve for a_1

$$a_{1}=20-50d$$

Steps for Solving Linear Equation

Multiply both sides of the equation by 2.

$$d\times 101^{2}+2\left(a_{1}-\frac{d}{2}\right)\times 101=4040$$

Calculate 101 to the power of 2 and get 10201.

$$d\times 10201+2\left(a_{1}-\frac{d}{2}\right)\times 101=4040$$

Multiply 2 and 101 to get 202.

$$d\times 10201+202\left(a_{1}-\frac{d}{2}\right)=4040$$

Use the distributive property to multiply 202 by a_{1}-\frac{d}{2}.

$$d\times 10201+202a_{1}+202\left(-\frac{d}{2}\right)=4040$$

Cancel out 2, the greatest common factor in 202 and 2.

$$d\times 10201+202a_{1}-101d=4040$$

Combine d\times 10201 and -101d to get 10100d.

$$10100d+202a_{1}=4040$$

Subtract 10100d from both sides.

$$202a_{1}=4040-10100d$$

Divide both sides by 202.

$$\frac{202a_{1}}{202}=\frac{4040-10100d}{202}$$

Dividing by 202 undoes the multiplication by 202.

$$a_{1}=\frac{4040-10100d}{202}$$

Divide 4040-10100d by 202.

$$a_{1}=20-50d$$

Solve for d

$$d=-\frac{a_{1}}{50}+\frac{2}{5}$$

Steps for Solving Linear Equation

Multiply both sides of the equation by 2.

$$d\times 101^{2}+2\left(a_{1}-\frac{d}{2}\right)\times 101=4040$$

Calculate 101 to the power of 2 and get 10201.

$$d\times 10201+2\left(a_{1}-\frac{d}{2}\right)\times 101=4040$$

Multiply 2 and 101 to get 202.

$$d\times 10201+202\left(a_{1}-\frac{d}{2}\right)=4040$$

Use the distributive property to multiply 202 by a_{1}-\frac{d}{2}.

$$d\times 10201+202a_{1}+202\left(-\frac{d}{2}\right)=4040$$

Cancel out 2, the greatest common factor in 202 and 2.

$$d\times 10201+202a_{1}-101d=4040$$

Combine d\times 10201 and -101d to get 10100d.

$$10100d+202a_{1}=4040$$

Subtract 202a_{1} from both sides.

$$10100d=4040-202a_{1}$$

Divide both sides by 10100.

$$\frac{10100d}{10100}=\frac{4040-202a_{1}}{10100}$$

Dividing by 10100 undoes the multiplication by 10100.

$$d=\frac{4040-202a_{1}}{10100}$$

Divide 4040-202a_{1} by 10100.

$$d=-\frac{a_{1}}{50}+\frac{2}{5}$$