A missile misses a target and goes into an orbit of 4R of geostationary orbit. Its period is 2days, 4days, 16

A missile misses a target and goes into an orbit of 4R of geostationary orbit. Its period is
2days, 4days, 16 days, 8 days

let's use v = d/t as t = d/v for the period

in orbit the centripetal force is equal to the gravitational force
take the geostationary one as basis for comparison
V1^2/r1 = g1
at 4R the g is roughly 1/16 g1 (inverse square rule)
so
v2^2/r2 = 1/16 v1^2/r1
and r2 = 4r1
v2^2/4r1 = 1/16 v1^2/r1
v2^2/v1^2 = 1/16 (4r1/r1) = 1/4
v2/v1 = 1/2
finally
the high orbit is 4 times lower one
and speed is half as fast
so
t = 4d/(1/2 v) = 8 t meaning 8 days

Centripetal force (Ms.Rω²) is provided by grav attraction (G.Ms.Me/R²)
Ms.Rω² = G.Ms.Me/R²
G.Me = R³ω² = constant (as G and Me are constants)

So .. R³ ∝ 1/ω² .. .. For a periodic time T .. ω = 2π/T, giving ..
R³ ∝ T² (Kepler's law)
T ∝ √R³
Changing R by a factor of 4 changes T by a factor .. √4³ = √64 = 8

Originally T = 1 day .. now increased by factor 8 .. .. ►8 days