(1)
an=(1/4)^n
bn+2=3log(an)
=3n
bn =3n-2
dn = an+bn
= 3n-2 + (1/4)^n
Sn = d1+d2+...+dn
= (3n-1)n/2 + (1/3)(1- (1/4)^n)
(2)
cn = an.bn
=(3n-2).(1/4)^n
let
f(x) =(3x-2).(1/4)^x
f'(x) = (1/4)^x .[ 3 - (3x-2)ln(1/4) ] =0
x = 1.388 ( max )
c1= 1/4
c2= 1/4
max cn = c1 or c2 = 1/4
cn≤(1/4)m^2+m-1
(1/4)m^2+m-1 ≥ 1/4
m^2+4m-5≥ 0
(m-1)(m+5)≥ 0
m≥ 1 or m≤-5