7.
$r = \frac{1}{r} + 1$
$r^2 - r - 1 = 0$
$\left( {r^{16} - r^{15} - r^{14} } \right) + (r^{15} - r^{14} - r^{13} ) + 2(r^{14} - r^{13} - r^{12} ) + 3(r^{13} - r^{12} - r^{11} ) + 5(r^{12} - r^{11} - r^{10} ) + 8(r^{11} - r^{10} - r^9 ) +$
$13(r^{10} - r^9 - r^8 ) + 21(r^9 - r^8 - r^7 ) + 34(r^8 - r^7 - r^6 ) + 55(r^7 - r^6 - r^5 ) + 89(r^6 - r^5 - r^4 ) + $
$144(r^5 - r^4 - r^3 ) + 233(r^4 - r^3 - r^2 ) + 377(r^3 - r^2 - r^1 ) + 610(r^2 - r - 1) = 0$
$r^{16} - 987r - 610 = 0$
ดังนั้น $r^{16} - 1 = 987r + 609$
$(r^8 - r^7 - r^6 ) + 3(r^7 - r^6 - r^5 ) + 4(r^6 - r^5 - r^4 ) + 7(r^5 - r^4 - r^3 ) + 11(r^4 - r^3 - r^2 ) + 18(r^3 - r^2 - r) + 29(r^2 - r - 1) = 0$
$r^8 + 2r^7 - 47r - 29 = 0$
ดังนั้น $r^8 + 2r^7 = 47r + 29$
จะได้ว่า $\frac{{r^{16} - 1}}{{r^8 + 2r^7 }} = \frac{{987r + 609}}{{47r + 29}} = \frac{{21(47r + 29)}}{{41r + 29}} = 21$