等比等差數列綜合題

24
Let the three number are a,ar,ar^2
Then a,ar+4,ar^2 is an arithmetic sequence
ar+4-a=ar^2-ar-4
8=ar(r-2)+a
On the other hand a,ar+4,ar^2+32 is a geometric sequence
So a(ar^2+32)=(ar+4)^2
4a=ar+2
a=2/(4-r)
So 4/[1+r(r-2)]=1/(4-r)
16-4r=1+r^2-2r
r^2+2r-15=0
r=3 or r=-5
if r=3, the three numbers are 2,6,18
if r=5, the three numbers are 2/9,-10/9,50/9
25 Want to find the ratio R of the sequence b_n
Let a_7=b_m, a_10=b_m+1, a_15=b_m+2
Let a_7=a, then a_10=a+3d, a_15=a+8d
So a(a+8d)=(a+3d)^2
a^2+8ad=a^2+6ad+9d^2
2ad=9d^2
2a=9d
As you know that R=b_m+1/b_m=1+3(d/a)=5/3
So b_n=3(5/3)^(n-1)