x 1 2 + x 4 + x 3 + x 2 + x + x 1 2 1 + x 4 1 + x 3 1 + x 2 1 + x 1 1
If x is a positive real number satisfying x 4 + x 4 1 = 1 9 4 , then find the value of the expression above.
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this is a highly overrated question.
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Yes, but I don't mind.
Problem solving is important.
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Shiva mann, I agree. That's why I like provide solutions to help others.
hey.. 7300802+2702+194+52+14+4 is not 7301066
I think You shouldn't put 2702 on it. because the question didn't ask you to sum x 6 + x 6 1
Your solution is good.
Given that x 4 + x 4 1 = 1 9 4 .
⇒ ( x 2 + x 2 1 ) 2 = 1 4 2 ⇒ x 2 + x 2 1 = 1 4 .
Now , from the conclusion that x 2 + x 2 1 = 1 4 , we will add 2 on both sides .
Now , ( x + x 1 ) 2 = 4 2 ⇒ x + x 1 = 4 .
x 3 + x 3 1 + 3 ( x + x 1 ) = ( x + x 1 ) 3 ⇒ x 3 + x 3 1 = 5 2 .
Squaring x 4 + x 4 1 = 1 9 4 on both sides ,We get,
x 8 + x 8 1 = 3 7 6 3 6 − 2 ⇒ x 8 + x 8 1 = 3 7 6 3 4 .
Now, ( x 8 + x 8 1 ) ( x 4 + x 4 1 ) = x 1 2 + x 1 2 1 + x 4 + x 4 1 .
Now, by substituting the respective values ,we get,
x 1 2 + x 1 2 1 = 7 3 0 0 8 0 2 .
Now,the given question is find x 1 2 + x 4 + x 3 + x 2 + x + x 1 2 1 + x 4 1 + x 3 1 + x 2 1 + x 1 ,
Which can easily be obtained by adding all the values in the boxes,that is 7 3 0 0 8 0 2 + 1 9 4 + 5 2 + 1 4 + 4 = 7 3 0 1 0 6 6 .
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It is given that x 4 + x 4 1 = 1 9 4 .
( x 2 + x 2 1 ) 2 = x 4 + 2 + x 4 1 = 1 9 6
⇒ x 2 + x 2 1 = 1 9 6 = 1 4
Similarly, x + x 1 = 1 4 + 2 = 4
( x 2 + x 2 1 ) ( x + x 1 ) = x 3 + x + x 1 + x 3 1
⇒ x 3 + x 3 1 = ( x 2 + x 2 1 ) ( x + x 1 ) − ( x + x 1 ) = 1 4 × 4 − 4 = 5 2
x 6 + x 6 1 = ( x 3 + x 3 1 ) 2 − 2 = 5 2 2 − 2 = 2 7 0 2
Similarly, x 1 2 + x 1 2 1 = 2 7 0 2 2 − 2 = 7 3 0 0 8 0 2
Therefore,
x 1 2 + x 4 + x 3 + x 2 + x + x 1 2 1 + x 4 1 + x 3 1 + x 2 1 + x 1
= 7 3 0 0 8 0 2 + 1 9 4 + 5 2 + 1 4 + 4 = 7 3 0 1 0 6 6