Application of fractional part in expansion

Algebra Level 5

\(\displaystyle (a+b\sqrt c)^{n}= I+f\) where \(\displaystyle I\) is integer part and \(\displaystyle f\) is fractional part..

Which of the following are true if \(\displaystyle a^{2}-1=b^{2}c\)

  1. Value of \(\displaystyle 1-f\) is \(\displaystyle I\)

  2. Value of \(\displaystyle 1-f\) is \(\displaystyle \frac { 1 }{ I } \)

  3. Expansion of \(\displaystyle (a-b\sqrt c)^{n}\) is \(\displaystyle 1-f\).

  4. Value of \(\displaystyle 1+f\) is \(\displaystyle I\)

  5. Value of \(\displaystyle 1+f\) is \(\displaystyle \frac { 1 }{ I } \)

  6. Expansion of \(\displaystyle (a-b\sqrt c)^{n}\) is \(\displaystyle 1+f\).

  7. \(\displaystyle \frac{1}{1-f}-f=I\)

  8. \(\displaystyle \frac{1}{1+f}-f=I\)

Submit the answer as the product of the true statements.

For example if 2,3,5 are true submit the answer as 30

\(\displaystyle 0 <a-b\sqrt c<1\) , n is a positive integer. a>1, c>0


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