Question

if (a + a^-1 = 2) find out (a^2020 + a^-2020 = ?)​

Answer 1

Answer:

your answer attached in the photo

Answer 2

$\displaystyle\huge\red{\underline{\underline{Solution}}}$

FORMULA TO BE IMPLEMENTED

We are aware of the identity that

${(x + y)}^{2} = {x}^{2} + 2xy + {y}^{2}$

GIVEN

$\displaystyle \: a + {a}^{ - 1} = 2$

TO DETERMINE

$\displaystyle \: {a}^{2020} + {a}^{ - 2020}$

EVALUATION

$\displaystyle \: a + {a}^{ - 1} = 2$

$\implies \: \displaystyle \: a + \frac{1}{a} = 2$

$\implies \: \displaystyle \: \frac{ {a}^{2} + 1}{a} = 2$

$\implies \: \displaystyle \: {a}^{2} + 1 = 2a$

$\implies \: \displaystyle \: {a}^{2} + 1 = 2a$

$\implies \: \displaystyle \: {a}^{2} - 2a + 1 = 0$

$\implies \: \displaystyle \: {(a - 1)}^{2} = 0$

$\implies \: \displaystyle \: a - 1 = 0$

$\implies \: a = 1$

Hence

$\displaystyle \: {a}^{2020} + {a}^{ - 2020}$

$= \displaystyle \: {(1)}^{2020} + {(1)}^{ - 2020}$

$= 1 + 1$

$= 2$