Question

if a+1/a =2 then a ^100+1/a100


plzzzz solve​

Answer 1

Solution :-

→ a + 1/a = 2

→ (a² + 1)/a = 2

→ a² + 1 = 2a

→ a² - 2a + 1 = 0

→ a² - a - a + 1 = 0

→ a(a - 1) - 1(a - 1) = 0

→ (a - 1)(a - 1) = 0

→ a = 1

Now,

→ a¹⁰⁰ + 1/a¹⁰⁰

→ 1¹⁰⁰ + 1/1¹⁰⁰

→ 1 + 1

→ 2 (Ans.)

______________________

Shortcut :-

→ a + 1/a = 2

Put a = 1

→ 1 + 1/1

→ 1 + 1

→ 2 (Satisfy).

Therefore,

→ a¹⁰⁰ + 1/a¹⁰⁰

→ 1¹⁰⁰ + 1/1¹⁰⁰

→ 1 + 1

→ 2 (Ans.)

______________________

Conclusion :-

• if a + 1/a = 2 , Put a = 1
• if a + 1/a = (-2) , put a = (-1)

_____________________________

Answer 2

${ \rm{ \huge{QUESTION }}}$




if a+1/a = 2 , then a^100 + 1/a^100 ???



${ \rm{ \huge{SOLUTION }}}$




${ \blue{ \tt{a + \frac{1}{a} = 2}}}$



${ \implies{ \blue{ \tt{ \frac{ {a}^{2} + 1 }{a} = 2}}}}$



${ \implies{ \tt{ \blue{ {a}^{2} + 1 = 2a}}}}$




${ \implies{ \blue{ \tt{ {a}^{2} - 2a + 1 = 0}}}}$




${ \implies{ \blue{ \tt{ {a}^{2} - (a + a) + 1 = 0}}}}$




${ \implies{ \blue{ \sf{ {a}^{2} - a - a + 1 = 0}}}}$



${ \implies{ \blue{ \tt{a(a - 1) - 1(a - 1) = 0}}}}$



${ \implies{ \blue{ \tt{(a - 1)(a - 1) = 0}}}}$




${ \implies{ \red{ \boxed{ \boxed{ \blue{ \tt{ \: \: \: a = 1 \: \: }}}}}}}$



Then,




${ \green{ \tt{ {a}^{100} + \frac{1}{ {a}^{100} } }}}$




${ \red{ \rightarrow{ \green{ \tt{ {1}^{100} + \frac{1}{ {1}^{100}}}}}} }$




${ \red{ \rightarrow{ \green{ \tt{1 + \frac{1}{1}}}}}}$



${ \red{ \rightarrow{ \green{ \tt{1 + 1 = 2}}}}}$



therefore,



${ \red{ \boxed{ \boxed{ \green{ \tt{ \: {a}^{100} + \frac{1}{ {a}^{100} } = 2 \: }}}}}}$