Question

if ${(8+3\sqrt{7})}^n=P+F$, where $P$ is an integer ad $F$ is a proper fraction then

• A. $P$ is an odd integer
• B. $P$ is an even integer
• C. $F(P+F)=1$
• D. $(1-F)(P+F)=1$

Answer 1

Answer: (A), (D)

$P$ is an odd integer
$(1-F)(P+F)=1$

Since $\because (8+3\sqrt{7})=P+F$ ...(1)
Here $0<F<1$
Let $F^{\prime }={(8-3\sqrt{7})}^n$ ...(2)
$0<F^{\prime }<I$
Adding (1) and (2)
$P+F+F^{\prime }={(8+3\sqrt{7})}^n+{(8-3\sqrt{7})}^n=2k$ (even integer) ..(3)
and $0<F+F^{\prime }<2\therefore F+F^{\prime }=I$
From (3) $P+1=2K,P=2K-1=$ odd integer
$\therefore (I-F)(P+F)=F^{\prime }(P+F)={(8-3\sqrt{7})}^n{(8+3\sqrt{7})}^n=I^n=1$