# 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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