Question

(X²⁰²⁰-1)÷(x-1) what is the answer for this question

Answer 1

ANSWER

${x}^{2019} + {x}^{2018} + {x}^{2017} + ......... + {x}^{2} + x + 1$

Steps

We have the formula

${a}^{n}  – {b}^{n}  = (a – b)( {a}^{n - 1}  + {a}^{n - 2} b + {a}^{n - 3} {b}^{2}  + ··· + a {b}^{n - 2}  + {b}^{n - 1} )$

So

${x}^{2020} - 1 \\ = {x}^{2020} - {1}^{2020} \\ = (x - 1)( {x}^{2019} + {x}^{2018} \times 1 + {x}^{2017} \times {1}^{2} + ......... + x \times {1}^{2018} + {1}^{2019}$

So,

$\frac{ {x}^{2020} - 1 }{x - 1} \\ = \frac{ (x - 1)( {x}^{2019} + {x}^{2018} \times 1 + {x}^{2017} \times {1}^{2} + ......... + x \times {1}^{2018} + {1}^{2019}}{x - 1} \\ = ( {x}^{2019} + {x}^{2018} \times 1 + {x}^{2017} \times {1}^{2} + ......... + x \times {1}^{2018} + {1}^{2019} \\ = ( {x}^{2019} + {x}^{2018} + {x}^{2017} + ......... + {x}^{2} + x + 1$

If x= 0 then the answer is

$\frac{ {0}^{2020} - 1}{0 - 1} \\ = \frac{ - 1}{ - 1} = 1$

If x = 1 then answer is

$\frac{ {1}^{2020} - 1 }{1 - 1} \\ = \frac{0}{0} = undefined$